Chapter 9 Quadratic Graphs Lesson 1: Graphing Quadratic Functions Lesson 2: Vertex Form & Shifts Lesson 3: Quadratic Modeling Lesson 4: Focus and Directrix Lesson 5: Equations of Circles and Systems Lesson 6: General Form of the Circle This assignment is a teacher-modified version of Algebra 2 Common Core Copyright (c) 2016 emath Instruction, LLC used by permission.
Chapter 9: Quadratic Graphs Lesson 1 Graphing Quadratic Functions Recall: A function in the form, where the leading coefficient a is not zero, is a quadratic function. A graph of a quadratic function is called a. The axis of symmetry of a parabola, is a line that divides the parabola into two halves. The axis of symmetry always passes through the of the parabola. The vertex of a parabola, also known as the, is where the graph has a or a value. Axis of Symmetry Equation: The axis of symmetry is the of the vertex. To find the y-coordinate, simply the x-value into the original equation.
A parabola will have a if the a value (leading coefficient) is and will have a if the a value is. 1.) Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function: y = 4x 2 + 8x + 3 (1) x = -1; (-1,15) (3) x = -1; (-1, -1) (2) x = 1; (1, 15) (4) x = 1; (1, 7) 2.) Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function. Then determine if the parabola has a maximum or minimum. y = -x 2-4x + 2
3.) Given the quadratic equation: y = x 2. (a) Create a table of values to plot this function over the domain interval -3 < x < 3. (b) Sketch a graph of this function on the grid to the right. (c) State the equation of the parabola's axis of symmetry. (d) State the coordinates of the turning point of this parabola. (e) Over what interval is this function increasing?
4.) Given the quadratic equation: f(x) = -x 2 + 6x + 5. (a) Graph this function on the grid to the right. (b) What is the range of this quadratic? (c) Over what interval is the graph of this function increasing? (d) Between what two consecutive integers does the larger solution to the equation -x 2 + 6x + 5 = 0 lie? Show this point on your graph.
5.) A sketch of the quadratic function y = x 2-11x - 26 is shown below marked with points at its intercepts and its turning point. Using your calculator, determine the coordinates for each of the points. The x-intercepts: A (Zeroes) B The y-intercept: D The turning point: C Over what interval is this function positive? 6.) At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t) = -16t 2 + 16t + 32, where s is measured in feet and t is measured in seconds. (a) At what time will the diver reach maximum height? (b) What is the diver s maximum height?
7.) This table shows the height in feet of some of the tallest building in the United States. The path of a model rocket can be modeled by the function: h(t) = -16t 2 + 100t If the rocket is launched from the ground, on which of the buildings on this list could the rocket land?
Chapter 9: Quadratic Graphs Lesson 1: Homework Graphing Quadratic Functions 1.) Which of the following represents the y-intercept of the graph of the quadratic function y = 2x 2 7x + 9? (1) 7 (2) 2 (3) -7 (4) 9 2.) For a particular quadratic function, the leading coefficient is negative and the function has a turning point whose coordinates are (-3,14). Which of the following must be the range of this quadratic? (1) {y y > -3} (2) {y y < -3} (3) {y y > 14} (4) {y y < 14} 3.) A parabola has one x-intercept of x = -2 and an axis of symmetry of x = 4. Which of the following represents its other x-intercept? (1) x = 3 (2) x = 10 (3) x = 6 (4) x = 8
4.) Given the quadratic equation: f(x) = x 2 + 2x 8. (a) Sketch a graph of f on the grid provided. y (b) Over what interval is f decreasing? x (c) Over what interval is f(x) < 0? (d) State the range of f. 5.) The number of meters above the ground, h, of a projectile fired at an initial velocity of 86 meters per second and at an initial height of 6.2 meters is given by h(t) -4.9t 2 + 86t + 6.2, where t represents the time, in seconds, since the projectile was fired. If the projectile hits its peak height at t = 8.775 seconds, which of the following is closest to its greatest height in meters? (1) 265 (2) 384 (3) 422 (4) 578
Chapter 9: Quadratic Graphs Lesson 2 Completing the Square & Shifting Parabolas The equation of a parabola can be expressed in either standard form,, or vertex form,. Vertex Form: where (h, k) is the x = h is the h represents the shift ( ) k represents the shift ( ) Exercise #1: The function y = x 2 is shown already graphed on the grid below. Consider the quadratic whose equation is y = x 2-8x + 18. (a) Using the method of completing the square, write this equation in the form y = (x - h) 2 + k.
Completing the Square: 1) move the constant to the other side of the equation (group the x s on one side) 2) Complete the square on x (take ½ of b and square it) 3) Add the number from step 2 to both sides of the equation 4) Factor the expression with x 5) Rewrite as a squared binomial 6) move constant back over to put into vertex form (b) Describe how the graph of y = x 2 would be shifted to produce the graph of y = x 2-8x + 18. (c) Sketch the graph of y = x 2-8x + 18 by using its vertex form in (a). What are the coordinates of its turning point (vertex)? Shifts of Quadratic Functions Given a quadratic function in vertex form y = (x h) 2 + k a shift from the basic graph y = x 2 is: (x h): shifts to the h units + k: shifts k units (x + h): shifts to the h units k: shifts k units
Exercise #2: Using your calculator and the window shown below, sketch the graphs of the simple quadratics. Every quadratic of the form y = ax 2 has a turning point at: Given the form of the quadratic equation y = a(x - h) 2 + k: For every a > 1, the parabola will be " " because it grows more. For every a < 1, the parabola will be " " because it grows more. Exercise #3: Place each of the following quadratic functions in vertex form and identify the turning point. (a) y = 3x 2 + 12x 2 (b) y = 2x 2 + 6x + 1
Exercise #4: If Charlotte completes the square for f(x) = x 2-12x + 7 in order to find the minimum, she must write f(x) in the general form f(x) = (x - a) 2 + b. What is the value of a for f(x)? (1) 6 (2) -6 (3) 12 (4) -12 Exercise #5: The function f(x) = 3x 2 + 12x + 11 can be written in vertex form as (1) f(x) = (3x + 6) 2-25 (3) f(x) = 3(x + 6) 2 25 (2) f(x) = 3(x + 2) 2-1 (4) f(x) = 3(x + 2) 2 + 7 Exercise #6: Find the turning point of the parabola by using any method. (a) (b)
Chapter 9: Quadratic Graphs Lesson 2: Homework Completing the Square & Shifting Parabolas 1.) The graph of y = x 2 is shown below. Which graph represents y = 2x 2? (1) (2) (3) (4) 2.) Which of the following equations would result from the shifting y = x 2 five units right and four units? (1) y = (x 5) 2 + 4 (2) y = (x + 5) 2 + 4 (3) y = (x 4) 2 5 (4) y = (x + 4) 2 5
3.) Which of the following represents the turning point of the parabola whose equation is y = (x + 3) 2 7? (1) (3, -7) (2) (-3, 7) (3) (-7, -3) (4) (-3, -7) 4.) Which of the following is a turning point of y = x 2 + 12x 4? (1) (12, -4) (2) (-6, -40) (3) (6, 104) (4) (-4, 12) 5.) Write each of the following quadratic functions in its vertex form by completing the square. Then, identify its turning point. (a) y = x 2 + 12x + 50 (b) y = -3x 2 + 30x + 7
Chapter 9: Quadratic Graphs Lesson 3 Quadratic Modeling Modeling with Quadratic Functions Quadratic functions are used to model many real life phenomenon. It is helpful to remember important features of a parabola when completing these problems. Recall: Max/Min: Axis of Symmetry: Exercise 1: Rachel tosses up an apple to her friend on a third-story balcony. After t seconds, the height of the apple in feet is given by the equation h = -16t 2 + 38.4t + 0.96. Her friend catches the apple just as it reaches its highest point. How long does the apple take to reach Rachel's friend? At what height above the ground did her friend catch it?
Exercise 2: An object is fired upwards with an initial velocity of 112 feet per second. Its height, in feet above the ground, as a function of time, in seconds since it was fired, is given by the equation: h(t) = -16t 2 + 112t. (a) At what height was the object fired? (b) Sketch a general curve of this equation below. (c) Algebraically, find the time that the rocket reaches its greatest height and the maximum height. Label these on the graph that you drew in part (b). (d) Algebraically, determine the time when the rocket reaches the ground. Label this on your graph in (b).
Exercise 3: The Crazy Carmel corn company has determined that the percentage of kernels that pop rises and then falls as the temperature of the oil the kernels are cooked in increases. It modeled this trend using the equation: (a) Algebraically determine the temperature at which the highest percentage of kernels pop. Also, determine the percent of kernels that pop at this temperature. (b) Using your calculator, sketch a curve below for P > 0. Label your window. (c) Using the ZERO command on your calculator determine, to the nearest degree, the two temperatures at which P = 0. Label them on your graph drawn in part (b). (d) If a typical batch of popcorn consists of 800 kernels, how many does the Crazy Carmel Corn company expect to pop at the optimal temperature?
(e) For a batch of popcorn to be successful, the company wants at least 85% of its kernels to pop. Write an inequality whose solution represents all temperatures that would ensure a successful batch. Solve this inequality graphically, to the nearest degree, and show your graph to below, labeling all relevant points. Exercise 4: A water balloon thrown vertically in the air reaches its peak height after 2 seconds. If its height, as a function of time, is given by h(t) = -16t 2 + bt + 6, then find the value of b.
Chapter 9: Quadratic Graphs Lesson 3: Homework Quadratic Modeling 1.) The height of a missile t seconds after it has been fired is given h = -4.9t 2 + 44.1t. Which of the following represents the number of seconds it will take for the rocket to reach its greatest height? (1) 108 (2) 4.5 (3) 99 (4) 7.5 2.) Find the number of units that produce the maximum revenue R = 900x 0.1x 2, where R is the total revenue (in dollars) and x is the number of units sold. 3.) A decathlete at the Olympics throws a javelin such that its height, h, above the ground can be modeled as a quadratic function of the horizontal distance, d, that it has traveled. Which of the following is a realistic quadratic function for this scenario? (1) (2) 1 75 3 (3) 100 2 h d d 1 75 3 (4) 100 2 h d d 1 75 3 100 2 h d d 1 75 3 100 2 h d d
4.) Bacteria tend to grow very fast in a Petri dish at first because of unlimited food and then begin to die out due to competition. In a certain culture, the number of bacteria is given by N(t) = -2t 2 + 92t + 625, where t represents the hours since 625 bacteria were introduced to the Petri dish. Determine the maximum number of bacteria that occur in the dish. 5.) A tennis ball is thrown upwards from the top of a 30-foot high building. Its height, in feet above the ground, t-seconds after it is thrown is given by h = -16t 2 + 80t + 30. (a) Algebraically determine the time when the tennis ball reaches its greatest height. What is that height? (b) The ball can be seen from the ground whenever it is at a height of at least 100 feet. Graphically determine the interval of time that the ball can be seen. Show your work by completing a sketch. Round your answers to the nearest hundredth.
Chapter 9: Quadratic Functions Lesson 4 Focus and Directrix The Locus Definition of a Parabola A parabola is the collection of all points from a fixed point (known as its ) and a fixed line (known as its ). Standard Form of Parabola: Vertex: Focus: Directrix: If p > 0, opens upward If p < 0, opens downward
Exercise #1: Given the equations of the parabolas below, identify the vertex, focus and directrix. (a) (x 2) 2 = 8(y 1) (b) (x + 1) 2 = -12(y + 2) (c) x 2 = -16y Exercise #2: Consider a parabola whose focus is the point (3, 1) and whose directrix is the line y = -3. Determine the equation of this parabola.
Exercise #3: Consider a parabola whose focus is the point (0, 7) and whose directrix is the line y = 3. (a) Sketch a diagram of the parabola below and identify its turning point. (b) Determine the equation of the parabola. Any line and any point not on the line when used as the focus and directrix define a parabola. The most challenging type of problem we will tackle in this course will be finding the equation of a parabola whose focus point is not on one of the two axes. We will, however, stick with horizontal lines as our directrices.
Exercise #4: Determine the equation of the parabola whose focus is the point (-3, 4) and whose directrix is the horizontal line y = 2. Exercise #5: Derive the analytic equation of the parabola shown in the diagram. The focus is (1,3) and directrix is y = 1.
Exercise #6: Regents Question Which equation represents the set of points equidistant from the line l and point R shown on the graph below? Exercise 7: The directrix of the parabola 12(y + 3) = (x 4) 2 has the equation y = -6. Find the coordinates of the focus of the parabola.
Chapter 9: Quadratic Functions Lesson 4: Homework Focus and Directrix 1.) Given the equations of the parabolas below, identify the vertex, focus, and directrix. (a) (x 3) 2 = -8(y 4) (b) (x + 4) 2 = -4(y + 1) (c) x 2 = 8y 2.) The parabola whose equation is is shown graphed on the grid below. Its directrix is the x-axis. (a) Explain why the focus must be the point (0,4). Label the point on the diagram. y x Directrix (b) Using the locus definition of a parabola, show that the equation is.
3.) Derive the analytic equation of the parabola shown in the diagram. Given the focus is (0,4) and directrix is y = 2. 4.) Derive the analytic equation of the parabola shown in the diagram.
Name Class Algebra II Common Core Practice: Locus of the Parabola (Focus, Vertex and Directrix) Identify the vertex, focus and directrix of each parabola below: 1. x 2 = 12y 2. x 2 = -16y 3. (x + 1) 2 = -8(y + 1) 4. (x + 2) 2 = 4(y + 1) 5. (x +2) 2 = -8(y + 2) 6. (x + 1) 2 = 4(y + 1)
7. (x 4) 2 = 4(y + 1) Write the equation of the parabola given the following conditions. 8. Focus (0, 15), directrix y = -15. 9. Focus (0, -25), directrix y = 25.
10. Focus (7, -1), directrix y = -9. 11. 12. 13.
Answer Key: 1.) Vertex: (0,0); Focus: (0,3); Directrix: y = -3 2.) Vertex: (0,0); Focus: (0,-4); Directrix: y = 4 3.) Vertex: (-1,-1); Focus: (-1,-3); Directrix: y = 1 4.) Vertex: (-2,-1); Focus: (-2,0); Directrix: y = -2 5.) Vertex: (-2,-2); Focus: (-2,-4); Directrix: y = 0 6.) Vertex: (-1,-1); Focus: (-1,0); Directrix: y = -2 7.) Vertex: (4,-1); Focus: (4,0); Directrix: y = -2 8.) x 2 = 60y 9.) x 2 = -100y 10.) (x 7) 2 = 16(y + 5) 11.) (x 3) 2 = 12(y + 2) 12.) (x + 2) 2 = 4(y + 2) 13.) (x 7) 2 = 4(y + 2)
Chapter 9: Quadratic Graphs Lesson 5 Equations of Circles and Systems The Equation of a Circle: A circle whose center is at and whose radius is given by: Exercise #1: Which of the following equations would have a center of (-3,6) and a radius of 3? (1) (x - 3) 2 + (y + 6) 2 = 9 (3) (x - 3) 2 + (y - 6) 2 = 3 (2) (x + 3) 2 + (y - 6) 2 = 9 (4) (x + 3) 2 + (y + 6) 2 = 3 Exercise #2: For each of the following equations of circles, determine both the circle's center and its radius. If its radius is not an integer, express it in decimal form rounded to the nearest tenth. (a) (x - 2) 2 + (y - 7) 2 = 100 (b) x 2 + y 2 = 121
(c) (x + 1) 2 + (y + 2) 2 = 1 (d) x 2 + (y - 3) 2 = 49 (e) (x + 6) 2 + (y - 5) 2 = 18 (f) (x - 4) 2 + (y - 2) 2 = 20 Exercise #3: For the following examples you are given the center of the circle, C and a point on the circumference of the circle, A. Determine an equation for each circle in center-radius form. (a) C: (-2, 1); A: (6, 4) (b) C: (0,-9); A(5,-5)
Exercise #4: Write equations for circles A and B shown below. Show how you arrive at your answers. Exercise #5: Solve the following system of equations graphically y = 3x - 5 x 2 + y 2 = 25 Recall: When graphing a circle, find the center and radius first; then graph the center and count out the radius to the left, right, up, & down from the center.
Exercise #6: Find the intersection of the circle (x + 2) 2 + y 2 = 36 and y = x - 4 algebraically. Recall: When solving a system algebraically, it helps to isolate one variable and then use substitution. In this problem, it is easier to get y alone for the linear equation and then substitute it in for y in the circle equation. Exercise #7: Solve the following system algebraically. (x 3) 2 + (y + 2) 2 = 16 2x + 2y = 10
Exercise #8: Solve the system graphically: x 2 + y 2 = 4 y = x 2 4
Chapter 9: Quadratic Graphs Lesson 5: Homework Equations of Circles and Systems 1.) What are the coordinates of the center and the length of the radius of the circle whose equation is (x + 1) 2 + (y 5) 2 = 16? (1) (1,-5) and 16 (2) (-1,5) and 16 (3) (1,-5) and 4 (4) (-1,5) and 4 2.) The equation of a circle is (x 3) 2 + y 2 = 8. The coordinates of its center and length of its radius are (1) (-3,0) and 4 (2) (3,0) and 4 (3) (-3,0) and (4) (3,0) and 3.) Which of the following represents the equation of the circle shown graphed below? (1) (x 2) 2 + (y + 3) 2 = 16 y (2) (x + 2) 2 + (y 3) 2 = 4 (3) (x 2) 2 + (y + 3) 2 = 4 x (4) (x + 2) 2 + (y 3) 2 = 16
4.) Circles are described below by the coordinates of their centers, C, and one point on their circumference, A. Determine an equation for each circle in center-radius form. (a) C(5,2) and A(11,10) (b) C(-2,-5) and A(3,-17) 5.) Find the intersection of the circle x 2 + y 2 = 29 and y = x 3 algebraically.
6.) Solve the following system of equations graphically. x 2 + y 2 = 25 y y = 5 x x 7.) Solve the following system of equations algebraically: (x 2) 2 + (y + 3) 2 = 4 X y = 3
Chapter 9: Quadratic Graphs Lesson 6 General Form of the Circle The General Form of the equation of a Circle is We can convert the general form of the equation of a circle to the standard form: (x h) 2 + (y k) 2 = r 2 We do so by on x and y. Exercise #1: By completing the square on both quadratic expressions in x and y determine the center and radius of a circle whose equation is Steps: x 2 + 10x + y 2 2y = 10 1) Group the x s and y s together 2) Move Constant to the other side. 3) Complete the square on x and y and add both numbers to both sides of the equation. 4) Factor the x and y expressions. 5) Rewrite factors as squared binomials.
Exercise #2: Identify the center and radius of the circles below by completing the square on x and y. (a) x 2 + y 2 + 10x - 4y + 4 = 0 (b) x 2 + y 2 + 2x + 6y + 1 = 0 Exercise #3: Convert the general form of the equation of a circle to the standard form (x h) 2 + (y k) 2 = r 2. (c) x 2 2x + y 2 15 = 0 (d)
Don't forget to take out the leading coefficient for x 2 and y 2 first before you can start the process of completing the square. (e) 4x 2 + 4y 2 + 12x + 4y + 1 = 0 (f) 5x 2 + 5y 2 20x 60y 45 = 0
Chapter 9: Quadratic Graphs Lesson 6: Homework General Form of the Circle 1.) By completing the square on each of the quadratic expressions, determine the center and radius of a circle whose equation is shown below. (a) x 2 6x + y 2 + 10y = 66 (b) x 2 + y 2 4x 6y 3 = 0 2.) The equation 4x 2 24x + 4y 2 + 72y = 76 is equivalent to (1) 4(x 3) 2 + 4(y + 9) 2 = 76 (3) 4(x 3) 2 + 4(y + 9) 2 = 166 (2) 4(x 3) 2 + 4(y + 9) 2 = 121 (4) 4(x 3) 2 + 4(y + 9) 2 = 436 3.) What are the coordinates of the center of a circle whose equation is x 2 + y 2 16x+ 6y+53 =0? (1) (-8,-3) (2) (-8,3) (3) (8,-3) (4) (8,3)
4.) Christopher s work for deriving the equation of a circle is shown below. In which step did he make an error in his work? (1) Step 1 (2) Step 2 (3) Step 3 (4) Step 4 5.) For the questions below, write the equation of circles in center-radius form. (a) x 2 + y 2 + 8x 2y + 13 = 0 (b) 3x 2 + 3y 2 + 18x + 6y + 18 = 0