Using the Laws of Exponents to Simplify Rational Exponents

Transcription

1 6. Explain Radicals and Rational Exponents - Notes Main Ideas/ Questions Essential Question: How do you simplify expressions with rational exponents? Notes/Examples What You Will Learn Evaluate and simplify expressions with rational exponents. Rewrite expressions in rational exponent form. Solve real-life problems involving rational exponents. Definition: An exponent that can be written as a What is a rational exponent? Rational Exponent Form: Practice: Using the calculator, evaluate the expressions Using the Laws of Exponents to Simplify Rational Exponents Practice: Simplify the expressions. Use only positive exponents in your answer. Multiply: Divide: Exponent to Exponent: ( ) 14. Find the area of the figure.

2 6. Explain Radicals and Rational Exponents - Notes Rewriting Expressions in Rational Exponent Form Examples: Property: Practice: Rewrite the expression in rational exponent form Practice: Simplify the expressions. Use only positive exponents in your answer. 1.. A 3. The radius r of a sphere is given by the equation r where A is the surface area of the 4 sphere. The surface area of a sphere is 1493 square meters. Find the radius of the sphere to the nearest tenth of a meter. Use 3.14 for Write the expression in exponential form. a + 3 a

3 6.3 Explain Exponential Functions - Notes Essential Question: What are some of the characteristics of the graph of an exponential function? Main Ideas/ Questions Notes/Examples What You Will Learn Identify and evaluate exponential functions. Graph exponential functions. Write exponential functions. Solve real-life problems involving exponential functions. In an Equation: A nonlinear function of the form: where,, and What is an exponential function? In a Table: As the independent variable changes by a constant amount, the dependent variable is multiplied by a constant factor (consecutive y-values form a constant ratio). Linear: Exponential Identifying and Evaluating Exponential Functions Practice: Determine whether the table represents a linear or an exponential function. Explain. Practice: Determine whether the equation represents an exponential function. Explain x y 51 x 3 7. y 7x y 6. Practice: Evaluate the function for the given value of x x 8. y 3; x 5 y x 10. y 4 ; x 3 x 34 ; x 4

4 6.3 Explain Exponential Functions - Notes Graphs of Exponential Functions y-intercept: (0, a) of the graph Asymptote: A line that the graph approaches but never intersects. The x-axis is an asymptote of the graphs above. Practice: Graph the function, identify the y-intercept and asymptotes of the graph, and describe the domain and range of the function x f x f x 1. x y-intercept: asymptote: Domain: Range: y-intercept: asymptote: Domain: Range: Writing Exponential Functions in Real-life Problems 13. The graph represents a bacterial population after days. a. Write an exponential function that represents the population. b. Identify and interpret the y-intercept. c. Describe the domain and range of the function. d. Find the population after 5 days. 14. The table shows the time (in minutes) since a cup of hot coffee was poured and the temperature (in degrees Fahrenheit) of the coffee. a. Use a graphing calculator to find an exponential function that fits the data. b. Predict the temperature of the coffee 10 minutes after it is poured.

5 6.4 Explain Exponential Growth and Decay - Notes Essential Question: What are some of the characteristics of exponential growth and exponential decay functions? Main Ideas/ Questions Notes/Examples What You Will Learn Use and identify exponential growth and decay functions. Interpret and rewrite exponential growth and decay functions. Solve real-life problems involving exponential growth and decay Exponential Growth and Decay Functions y = ab x Exponential Growth Functions In the form y = ab x, where b is replaced by (1 + r). Exponential Decay Functions In the form y = ab x, where b is replaced by (1 r). How can you determine if an exponential function is a growth or decay function? Practice: Identify the function as a growth or decay function, the initial amount a, the rate of growth or decay r (as a percent), and evaluate the function when x = 5. Round your answer to the nearest hundredths place. 1. y = 350(1 +.75) x. y = 575(1 0.6) x 3. f(t) = 475(0.5) x Growth/Decay Growth/Decay Growth/Decay a = a = a = r = r = r = When x = 5, y When x = 5, y When x = 5, f(x) 4. h(x) = 1(1.05) x 5. p(x) = 1.8 x 6. Growth/Decay Growth/Decay Growth/Decay a = a = a = r = r = r = When x = 5, h(x) When x = 5,p(x)

6 6.4 Explain Exponential Growth and Decay - Notes Steps. Step 1: Is it an Exponential Growth/Exponential Decay? Step : Identify a, the initial amount Step 3: Identify r, the rate of growth or decay (in decimal form) Step 3: Identify b, the growth factor (1 + r) or the decay factor (1 r) Practice: Write a function that represents the situation. 7. In 005, there were 100 rabbits in Polygon Park. The population increased by 11% each year. Growth/Decay a = r = Equation: b = What will the population be in 05? (Round to the nearest whole number) Writing and Interpreting Exponential Growth and Decay Functions 8. A $900 sound system decreases in value by 9% each year. Growth/Decay a = r = b = Equation: 9. The table shows the balance of a money market account over time. Write a function that represents the balance after t years. Growth/Decay Equation: a = r = b = 10. The function y = 4(0.50) x represents the number y of teams left in a baseball tournament after x rounds. a. Determine whether the function represents exponential growth or exponential decay. b. Identify the initial amount. c. Identify and interpret the growth factor or decay factor. 11. The value of a car is $1,500. It loses 1% of its value every year. (a) Write a function that represents the value y (in dollars) of the car after x years. (b) Graph the function. (c) Identify and interpret any asymptotes of the graph. (d) Estimate the value of the car after 6 years.

7 6.5 and 6.6 Explain Geometric Sequences - Notes Essential Question: How can you use a geometric sequence to describe a pattern? How can you define a sequence recursively? Main Ideas/ Questions What You Will Learn Notes/Examples Identify and graph geometric sequences. Write geometric sequences as functions. Write terms of recursively defined sequences. Vocabulary Sequence: an ordered list of numbers Term (a n ): each number in the sequence **Each term has a specific position n in the sequence. Geometric Sequence: The ratio between each pair of consecutive terms is the same. The ratio is called the common ratio, r. Example: 40, 0, 10, 5,..a n. r = a 1 = a 3 = a = a 4 = Practice: Determine whether each sequence is a geometric sequence. If it is a geometric sequence, then find the common ratio. If it is not, then explain why.,,,,.... 3, 7, 11, 15, 3. Yes, r = Yes, r = Yes, r = No, because No, because No, because 4. Write the next three terms of a geometric sequence where a 1 = and r = 1. To Graph.. Let a term s position number n in the sequence be the x-value. The term a n is the corresponding y-value. Plot the ordered pairs (n, a n ). Graphing Sequences 5. Graph the geometric sequence 6, 1, 4, 48, a. Make a table: b. What do you notice?

8 6.5 and 6.6 Explain Geometric Sequences - Notes Equation for a Geometric Sequence a n = a 1 r n 1 (The equation above is in the form, and is an explicit rule (can find any term in a sequence.) Practice: Write an equation for the nth term of each geometric sequence. Then find a 6. (Hint: Find "r" first and a 1, the first term.) 6. 3, 15, 75, 375, a 1 = a 1 = a 1 = r = r = r = Equation: Equation: a 6 = a 6 = Equation: a 6 = 9. Clicking the zoom-out button on a mapping website doubles the side length of the square map. a. Explain why this is an example of a geometric sequence. Zoom-out clicks, n 1 3 Map side length (miles), f(n) b. Write a function that represents the sequence. c. After how many clicks on the zoom-out button is the side length of the map 640 miles? Recursively Defined Sequences ***When writing a recursive rule for a sequence, you must write the beginning term(s) and the recursive equation.*** Practice: Write the first three terms of the sequence. 10. a1 ; an an a1 4; an 3an 1 a n = r a n 1 where r is the common ratio Practice: The function f represents a sequence. Find the nd, 5 th, and 10 th terms of the sequence. 1. f(1) = 8, f(n) = f(n 1) 13. f(1) = 1, f(n) = f(n 1)

9 4.7 and 6.6 Explain Arithmetic Sequences - Notes Essential Question: How can you use an arithmetic sequence to describe a pattern? How can you define a sequence recursively? Main Ideas/ Questions What You Will Learn Notes/Examples To write and graph the terms of arithmetic sequences. To write arithmetic sequences as functions. Write terms of recursively defined sequences. Write recursive rules for sequences. Arithmetic Sequences The difference between each pair of consecutive terms is the same The difference is called the common difference, d. 5, 1, -3, -7,.. Geometric Sequences The ratio between each pair of consecutive terms is the same The ratio is called the common ratio, r. 1, 5, 5, 15, Practice: Determine if the following sequences are an example of an arithmetic sequence, a geometric sequence, or neither. If it is an arithmetic sequence, then find the common difference, d. If it is a geometric sequence, then find the common ratio, r.,,,,.... 1,, 4, 7, , -3, -9, -7, Arithmetic, d = Arithmetic, d = Arithmetic, d = Arithmetic, d = Geometric r = Geometric, r = Geometric r = Geometric, r = Neither Neither Neither Neither 5. Write the next three terms of the sequence:, 7, 1, 176, Write the next three terms of the sequence: -0, -30, -45, -67.5,.. To Graph.. Let a term s position number n in the sequence be the x-value. The term a n is the corresponding y-value. Plot the ordered pairs (n, a n ). Graphing Sequences 7. Does the graph represent an arithmetic sequence? a. Make a table: b. Yes or No? Explain c. Compare the common difference and the slope.

10 4.7 and 6.6 Explain Arithmetic Sequences - Notes Writing Arithmetic Sequences as Functions 8. Given: 5, 8, 11, 14, 17,.. a. What is the common difference? d = b. Fill in the table. Position, n Process Term, a n a 1 = a = a 3 = a 4 = a 5 = 9. Formula for finding the nth term: Practice: Write an equation for the nth term of each arithmetic sequence. Then find a 50. (Hint: Find "d" first!!) , 11, 8, 5, , 0, -1, -,. a n = + ( - 1) _ Simplify: a 50 = 1. Online bidding for a purse increases by $5 for each bid after the $60 initial bid. a. Arithmetic/Geometric/ Neither? Bid number, n 1 3 Bid amount f(n) b. Write a function that represents the sequence. c. The winning bid is $105. How many bids were there? Recursively Defined Sequences Practice: Determine whether the recursive rule represents an arithmetic sequence or a geometric sequence. Then identify the d or the r depending on what type of sequence it is. Write the first three terms of the sequence. a n = a n 1 + d 13. a 1 = 18, a n = a n Arithmetic, d = Geometric r = where d is the common difference Practice: The function f represents a sequence. Find the nd, 5 th, and 10 th terms of the sequence. 14. f(1) = 8, f(n) = f(n 1) f(1) = 1, f(n) = f(n 1) + 4

11 7. Explain Multiplying Polynomials - Notes Main Ideas/ Questions What You Will Learn Essential Question: How can you multiply two polynomials? Notes/Examples Rewrite polynomial expressions of degree one and two using the distributive property. Multiply binomials and trinomials. Polynomial A number, variable, or the product of a number and one or more variables with whole number exponents (can be a sum of the above) What do you already know? How to Distribute..multiply coefficients and add exponents if the bases are the same How to Combine Like Terms Add or Subtract the Coefficient, do NOTHING with the exponent Multiplying Polynomials Using the Distributive Property Binomial: a polynomial with terms Trinomial: a polynomial with 3 terms Practice: Write equivalent expressions. 1. 3x(x 5) + 6x. 3m ( 3 m + 3) + (5 m 1) 3. 1x 4x( 3x ) + x 3 1 st Rewrite: Split the first binomial and repeat the next. Practice: Use the distributive property to find the product. 4. x x 1 5. b 3b nd Simplify by distributing and combining like terms Example:

12 7. Explain Multiplying Polynomials - Notes 6. a 1 a n 4n 1 8. (x + 5) 10. 3n n 5n 1 9. d 3d 4d Application: Write a polynomial that represents the area of the shaded region. Making Connections Whenever you multiply the binomials, you are writing equivalent expressions. Explain how to check your answer using your calculator.

13 7.4 Explain Dividing Polynomials - Notes Main Ideas/ Questions Essential Question: How can you divide two polynomials? Notes/Examples What You Will Learn Divide polynomials by monomials. Use polynomial long division to divide polynomials by other polynomials. What do you already know? How to Divide Powers with the Same Base divide coefficients and subtract exponents when the bases are the same. How to Combine Like Terms Add or Subtract the Coefficient, do NOTHING with the exponent. How to Divide Using Long Division: Divide, Multiply, Subtract (Take the opposite of the entire polynomial), Bring down, Repeat if needed Write your final solution as quotient + remainder, if necessary. divisor ) If there is a remainder, then If there is not a remainder, then. How can you check your work? Dividing Polynomials by Polynomials Steps: Always Arrange terms in form. (insert 0 as a coefficient for missing terms) Then Divide, Multiply, Subtract, Bring down the next term. Repeat steps -5 as necessary. Practice: Use polynomial long division to divide. Determine whether the divisor is a factor (divides evenly into the dividend).. (0x + 5x) 5x Answer: Is 5x a factor? Explain.

14 7.4 Explain Dividing Polynomials - Notes 3. (4x + 5) (x + 1) 4. (y + 3y 40) (y + 5) Answer: Answer: 5. (b b 3) (b + 1) 6. (7x + 3x 4) (x x + 5) Answer: Is (b + 1) a factor? Explain. Answer: Is (x x + 5) a factor? Explain. Practice: Write the following polynomials in standard form (add a 0 as a coefficient for missing terms) and divide. 7. (4b 3) (b 1) Answer: Application 8. The area of a rectangle is represented by the expression 1x 18x 1. The length of the rectangle is 6x 3. Write an expression that represents the width of the rectangle.

15 7.5 Explain Solving Polynomial Equations in Factored Form - Notes Main Ideas/ Questions What You Will Learn Essential Question: How can you solve a polynomial equation? Notes/Examples Use the Zero-Product Property. Factor polynomials using the GCF. Use the Zero-Product Property to solve real-life problems. Vocabulary Factored Form: A polynomial that is written as a product of factors. Standard Form: Factored Form: ( ) ( )( ) Roots: Solutions to polynomial equations. When equations are written in factored form and equal to 0 we can easily solve the equations to find the roots. Steps to Solve Polynomials in Factored Form Using the Zero-Product Property Step 1: Set equation equal to 0. Step : Set each factor equal to 0 and solve. Solving Equations Written in Factored Form Practice: Solve the equations that are written in factored form. 1. xx 5 0. c c b b 4. g6 3g6 3g 0 5. x Use the Area the GCF Model to Identifying (the largest Remove term that divides evenly into each term) the GCF Solving Equations that are Not in Factored Form Factoring Using the GCF Why do we factor? Used to write polynomials in factored form. 6. and + 7. and + 8. and + ***When finding the GCF of coefficients look for the _ factor. ***When finding the GCF of variables look for the _ exponent.

16 7.5 Explain Solving Polynomial Equations in Factored Form - Notes Steps: Practice: Identify Use the the area GCF. model Divide to remove each term the greatest by the GCF. common factor Factoring Using the GCF Practice: Factor the polynomial x 3x 10. 4y 0y u 6u _ ( ) Solving Equations by Factoring Steps to Solve Equations Using the Zero-Product Property Step 1: Set equation equal to 0. Step : If not in factor form, factor using the GCF. Step 3: Set each factor equal to 0 and solve Practice: Solve the equation. (Find the roots.) 1. 35n 49n z 5z x 15. s 11s 6x Real-life: A boy kicks a ball in the air. The height y (in feet) above the ground of the ball is modeled by the equation y 16x 80 x, where x is the time (in seconds) since the ball was kicked. Find the roots of the equation when y 0. Explain what the roots mean in this situation.

17 7.6 and 7.7 Explain Day 1 Factoring ax + bx + c - Notes Main Ideas/ Questions Essential Question: How do you factor a polynomial in the form ax + bx + c? Notes/Examples What You Will Learn Factor ax + bx + c. Use factoring to solve real-life problems. Vocabulary Trinomial: polynomial that has three terms.standard Form ax + bx + c = 0 Factoring by Grouping Used when. factoring a polynomial with 4 terms and is needed sometimes when factoring trinomials. Steps: Step 1: Put the four terms into an area model. Step : Find the GCF for the top row and the bottom row. Step 3: Verify the outsides of the puzzle work. If not, rearrange the order of the terms in the model. 1. 8s 3 + s + 64s h 18h 35h a a 30a 5 Factoring Trinomials Used with Trinomials in the Form: ax + bx + c Consider the following area model and resulting 4. Factor the trinomial by completing the area model. trinomial: x + 8x + 15 (x + ) (x + 4) = x + 6x + 8 x x x x 4 4x 8 1) What calculation created the 8? x 15 ) What calculation created the 6x?

18 7.6 and 7.7 Explain Day 1 Factoring ax + bx + c - Notes Practice: Factor the trinomials by completing the area model. 5. a + 1a y 4y 3 7. x + 3x 18 a 7 y -3 x -18 Practice: Factor the trinomials by completing the area model. 8. c + 5c x 10x b + 7b 18 Goal: to write the polynomial in factored form. Step 1: Factor out the GCF Step : Look at the resulting trinomial. Can you factor again using the area model? 11. c 14c a 8a x + 1xy 54y Factoring Completely

19 7.6 and 7.7 Explain Day Factoring ax + bx + c - Notes Essential Question: How do you factor a polynomial in the form ax + bx + c? Main Ideas/ Questions What You Will Learn Notes/Examples Factoring with ax + bx + c when there is a leading coefficient for x. 1. Factor the trinomial by completing the area model. y 9y + 14 Factoring Trinomials Continue to consider the structure. Consider the following area model and resulting trinomial: y 14 (3x ) (x + 4) = 3x + 10x 8 3x - x 3x -x 1) What calculation created the -9y? 4 1x -8 1) What calculation created the 10x? ) What calculation created the 14? ) What calculation created the -8? 3) Where could the -4 calculation come from? Factoring Trinomials where the a is a prime number There is only one possibility for the factors of the squared term Practice: Factor the trinomials by completing the area model. 3. 5n + 17n x + 13x a 3a 14 5n 5n x x a a

20 7.6 and 7.7 Explain Day Factoring ax + bx + c - Notes Practice: Factor the trinomials by completing the area model. 6. y + 15y x 11x b + 4b 7 Factoring Trinomials where the a is NOT a prime number There are multiple possibilities for the factors of the squared term 9. 9y + 6y a + 8a x 7x 0 9y x + 13x y + 7y + 1 Factoring Completely Beware Sometimes expressions can t be factored!!!!! 9. 9x x 11x +

21 7.8 Explain Difference of Two Squares and Solving Equations - Notes Essential Question: How do you factor the difference of two squares? How do you solve equations? Main Ideas/ Questions Notes/Examples What You Will Learn Factor the difference of two squares. Use factoring to solve real-life problems Perfect Squares 1 = _ = _ 3 = _ 4 = _ 5 = _ 6 = _ 7 = _ 8 = _ 9 = _ 10 = _ Factoring the Difference of Two Squares Consider the structure of the area model and answer Practice: Factor the polynomial Pay attention to the structure of the answers. 1. s 49. t x a 9 a 3 (a + 3)(a 3) 4. 4g h k Solving Equations by Factoring Steps to Solve Equations Using the Zero-Product Property Step 1: Set equation equal to 0. Step : If not in factor form, factor using the GCF and/or the area model as needed. Step 3: Set each factor equal to 0 and solve. Practice: Factor and then solve the equation. 7. g 6g = w = 30w

22 7.8 Explain Difference of Two Squares and Solving Equations - Notes Practice: Factor and then solve the equation. 9. r 3r = 10. 6x 11k + 4 = 0 Practice: Factor and then solve the equation. 11. x y The area of a right triangle is 16 square miles. One leg of the triangle is 4 miles longer than the other leg. Find the length of each leg. Real-Life 14. The length of a rectangular shaped park is (3x+5) miles. The width is (x+8) miles. The area of the park is 360 square miles. What are the dimensions of the park?

23 7.9 Explain Factoring Completely - Notes Main Ideas/ Questions Essential Question: How do you factor a polynomial completely? Notes/Examples What You Will Learn Factor completely. Use factoring to solve real-life problems. Factoring Completely Goal: to write the polynomial in fully factored form. To factor completely you should try each of these steps: Step 1: Factor out the GCF. Step : Determine if you can factor further using the area model. Reminder: ALWAYS write the GCF in front of the final answer.don t forget In Exercises 1-4, factor the polynomial completely. 1. 4c 4c 3. a 3a 3. 0 p p 1 4. x + 6x 16

24 7.9 Explain Factoring Completely - Notes Steps to Solve Equations Using the Zero-Product Property Step 1: Set equation equal to 0. Step : If not in factor form, factor using the area model. Step 3: Set each factor equal to 0 and solve. In Exercises 5 9, solve the equation by factoring. 5. 3x 1x c 108 = y + 50y = = 3n 9. (3c )(c + ) = The sum of the first n consecutive even numbers can be found using S = n + n, where n. What is the value of n when the sum is 156? F 6 G 39 H 6 J 1

25 Explain - Intro to Quadratic Function Notes Essential Question: What are the characteristics of the graph of a quadratic function? Main Ideas/ Questions What You Will Learn Notes/Examples Identify the vertex, axis of symmetry, zero, roots, x-intercept, y-intercept, and maximum/minimum value of quadratic functions. Find the domain and range of quadratic functions. Define, Draw and Label: Vertex A Quadratic Function makes a U shaped graph called a Parabola y-intercept _ x-intercepts Axis of Symmetry _ What is the Vertex? ( ) What is the y-intercept? ( ) What are the x-intercepts? ( ) ( ) What is the Axis of Symmetry (equation) _ What is the Domain: What is the Range Does it have a Maximum or Minimum Value? (circle) Does it open Up or Down (circle) MINIMUM Parabola the vertex is at the bottom MAXIMUM Parabola the vertex is at the top A quadratic can have. _,, or _ solutions. Solutions are also called: x-intercepts, roots, or zeros.

26 Practice: Identify the characteristics of the quadratic function and its graph and label them clearly on the graph. 1. h(x) = x + 8 Vertex: Opens: Up or Down Axis of Symmetry: Maximum Value: or Minimum Value: y-intercept: x-intercept(s): Domain: Range:. f(x) = x 8x + 8 Vertex: Opens: Up or Down Axis of Symmetry: Maximum Value: or Minimum Value: y-intercept: solution(s): Domain: Range: 3. f(x) = x 4x + 6 Vertex: Opens: Up or Down Axis of Symmetry: Maximum Value: or Minimum Value: y-intercept: zero(s): Domain: Range: 4. y = 3x + 6x + Vertex: Opens: Up or Down Axis of Symmetry: Maximum Value: or Minimum Value: y-intercept: Solution(s): Domain: Range:

27 8.1 Explain Graphing f(x) = ax - Notes Main Ideas/ Questions What You Will Learn Essential Question: How does the value of a affect the graph of f(x) = ax? Notes/Examples Graph quadratic functions of the form f(x) = ax Compare and contrast quadratic functions of the form f(x) = ax to the parent function, f(x) = x. Equation: y a = What is the quadratic parent function? Graph: x Vertex: Axis of Symmetry: y-intercept: Opens: Up or Down Maximum Value: or Minimum Value: x-intercept(s): Domain: Range: Dilations - f(x) = ax Practice: Graph the function. Compare the graph to the graph of f(x) = x, where a = _ & Vertex: 1. g(x) = x a = _. m(x) = 1 5 x a = _ Vertex: Vertex: If only the a value is changed, does it affect the location of the vertex? If a is positive, the parabola opens _. If a is negative, the parabola opens _. ****Once you have used the negative sign, if there is one, you are finished with it**** If a > 1, without its sign, If 0 < a < 1, without its sign, Practice: Describe the transformations for #1 and # 1..

28 8.1 Explain Graphing f(x) = ax - Notes Dilations - f(x) = (ax) - Practice: Graph the function. Compare the graph to the graph of f(x) = x, where a = _ & Vertex: 3. m(x) = (x) - 4. n(x) = -( 1 5 x) a = _ a = _ Vertex: Vertex: If a is positive, the parabola opens up. If a is negative, the parabola opens down. ****Once you have used the negative sign, if there is one, you are finished with it**** If a > 1, without its sign, If 0 < a < 1, without its sign, Practice: Describe the transformations for #3 and # A Once vertical stretch you understand has the same visual what affect the as a horizontal numbers shrink/compression. do to a graph, you A vertical can shrink/compression describe the has transformation the same visual affect without as a horizontal ever stretch. looking y = 9x and at y = the (3x) graph. Practice: Write an equation that would Reflections result in the described transformation from the parent function: 5. A ****If parabola there that is a opens negative downward in front and of x, inside or outside 6. A parabola the parenthesis, that opens then upward a reflection and has a has a vertical stretch. occurred**** vertical shrink/compression. 5. y = x 6. y = ( x) Outside: Reflection in the 7. A parabola that opens downward and has a vertical shrink/compression. Inside: Reflection in the.. 8. A parabola that opens upward and has a vertical stretch.