Study Resources For Algebra I Unit 2A Graphs of Quadratic Functions This unit examines the graphical behavior of quadratic functions. Information compiled and written by Ellen Mangels, Cockeysville Middle School August 2014 Algebra I Page 1 Unit 2A. Graphs of Quadratic Functions
Topics: Each of the topics listed below link directly to their page. First Differences Second Differences Intervals o Of Increase o Of Decrease o Interval Notation Example of Increasing or Decreasing Intervals Domain and Range X and Y intercepts Zeros o Finding the Zero on the Graphing Calculator Roots of a Quadratic Function Absolute Maximum Absolute Minimum Vertex Axis of Symmetry Interval Notation Average Rate of Change Standard Form Factored Form Vertex Form Transformations o Translation Vertical Horizontal o Reflection Vertical Horizontal o Dilation Stretch Shrink o Putting it all together Algebra I Page 2 Unit 2A. Graphs of Quadratic Functions
Algebra I Page 3 Unit 2A. Graphs of Quadratic Functions
First Differences A look at how to calculate First Differences: http://www.youtube.com/watch?v=cr1lkgo7hna x y 0 0 1 30 2 60 3 90 4 120 + 30 + 30 + 30 + 30 The first difference is 30. The first difference is constant. Therefore, the table of values represents a linear function of the form y = mx + b. x y 0 0 1 1 2 4 3 9 4 16 + 3 + 5 + 7 The first difference is not constant. Therefore, the table of values does not represent a linear function. Algebra I Page 4 Unit 2A. Graphs of Quadratic Functions
Second Differences x y 0 0 1 1 2 4 3 9 4 16 The second difference is 2. + 3 + 5 + 7 + 2 + 2 + 2 The second difference is constant. Therefore, the table of values represents a quadratic function of the form y = ax 2 + bx + c (Source: http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/transformationhirev1.shtml) Algebra I Page 5 Unit 2A. Graphs of Quadratic Functions
Looking at Second Differences Example: y = x 2 4x x y 0 0 1-3 2-4 3-3 4 0-3 - 1 + 3 The second difference is +2. + 2 + 2 + 2 (Source: http://mathbitsnotebook.com/algebra1/quadratics/qdgraphinfo.html) When the second difference is positive, the parabola opens up. Algebra I Page 6 Unit 2A. Graphs of Quadratic Functions
Looking at Second Differences Example: y = -x 2 + 4x x y 0 0 1-5 2-12 3-21 4-32 - 5-7 - 9-11 The second difference is -2. - 2-2 - 2 (Source: http://mathbitsnotebook.com/algebra1/quadratics/qdgraphinfo.html) When the second difference is negative, the parabola opens down. Algebra I Page 7 Unit 2A. Graphs of Quadratic Functions
Intervals of Increase (Source: http://www.mathsisfun.com/sets/functions-increasing.html) Algebra I Page 8 Unit 2A. Graphs of Quadratic Functions
Intervals of Decrease (Source: http://www.mathsisfun.com/sets/functions-increasing.html) Interval Notation Interval Notation: [-1, 3) (Source: http://catalog.flatworldknowledge.com/bookhub/reader/128?e=fwk-redden-ch02_s07) Algebra I Page 9 Unit 2A. Graphs of Quadratic Functions
Example of Increasing or Decreasing Intervals This function is decreasing over the interval: 0 x 2 In other words, the graph is going downhill starting at x= 0 until it gets to x = 2. This function is increasing over the interval: 2 x < In other words, the graph is going uphill starting at x= 2 and continues to increase forever. Algebra I Page 10 Unit 2A. Graphs of Quadratic Functions
Domain and Range (Source: http://www.mathsisfun.com/definitions/domain-of-a-function.html) (Source: http://www.mathsisfun.com/definitions/range-of-a-function.html) Video How do I find the domain and range on a graph of a function? http://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-from-graphs Algebra I Page 11 Unit 2A. Graphs of Quadratic Functions
x and y intercepts What is an x-intercept? What is a y-intercept? Simply put, these are the points on the x and y axis where the quadratic function crosses the x and y axis. x-intercepts y-intercept Most parabolas will intercept the x-axis at two points. Sometimes the parabola will intercept the x-axis only once or not at all. Algebra I Page 12 Unit 2A. Graphs of Quadratic Functions
Zeros of a Quadratic Function Remember the x-intercept? An x-intercept is also called the Zero of a function. Why? Because when the value of function is zero, the point is on the x-axis. (-4, 0) (1, 0) The zero of a function is also sometimes referred to as the Root of the Function. Algebra I Page 13 Unit 2A. Graphs of Quadratic Functions
Finding the Zero of a Quadratic Function on the TI-84 Example: y = - 1 2 x2 + 2x + 6 Step 1: Type the equation into Y= Step 2: Press GRAPH. In order for the next step to work, you have to be able to see the zeros of the function. If you can t, adjust the x and y axis by pressing WINDOW and changing the values for Xmin, Xmax, Ymin, Ymax. Step 3: Press 2 ND TRACE (CALC) Select 2: zero Step 4: The calulator will now ask you three quesitons: Left Bound? Right Bound? Guess? Use the left arrow to move the blinking spider to the left of the zero. Press ENTER. Use the right arrow to move the blinking spider to the right of the zero. Press ENTER. You should notice a small triangle at the top of the screen pointing towards the zero. When you get to the Guess question, you can move the blinking spider to where you think the zero will be, but you really don t have to. Just press ENTER again. Congratulations! You have just found your first zero of a quadratic function (-2, 0). Now you get to do steps 3 and 4 again to find the zero on the other side of the parabola. Try that one on your own. You should get the point (6, 0). Note: Sometimes due to rounding issues with the graphing calculator, you may get a value for y such as 1E-12 instead of zero. Don t panic. This value is 0.000000000001 which is so close to zero we are going to call it zero! Algebra I Page 14 Unit 2A. Graphs of Quadratic Functions
Absolute Maximum of a Quadratic Function The highest point of a parabola that opens downward. max Absolute Minimum of a Quadratic Function The lowest point of a parabola that opens upward. min Algebra I Page 15 Unit 2A. Graphs of Quadratic Functions
Vertex of a Quadratic Function The vertex is the absolute maximum or minimum point of a parabola. vertex vertex The vertex is (2.5, 1). The vertex is (-1.5, -3) Algebra I Page 16 Unit 2A. Graphs of Quadratic Functions
The Axis of Symmetry of a Quadratic Function The Axis of Symmetry is the vertical line that passes through the vertex of the parabola and divides the parabola into two symmetrical halves. The equation for the Axis of Symmetry is x = 2.5 The equation for the Axis of Symmetry is x = -1.5 The equation for the Axis of Symmetry will always be written in the form of X= for a Quadratic Function because that is how we write equations for vertical lines. Every point on a vertical line has the same x value. Notice that the equation for the Axis of Symmetry uses the same value as the x-value of the vertex. (see the previous page) Algebra I Page 17 Unit 2A. Graphs of Quadratic Functions
Interval Notation Open Interval Example: 2 < x < 5 written in interval notation: ( 2, 5 ) 0 1 2 3 4 5 6 7 Closed Interval Example: 2 x 5 written in interval notation: [ 2, 5 ] 0 1 2 3 4 5 6 7 Half Closed or Half Open Interval Example: 2 x < 5 written in interval notation: [ 2, 5 ) 0 1 2 3 4 5 6 7 Example: 2 < x 5 written in interval notation: ( 2, 5 ] 0 1 2 3 4 5 6 7 Algebra I Page 18 Unit 2A. Graphs of Quadratic Functions
Interval Notation with Quadratic Functions 2 This function is decreasing over the interval: 0 x 2 Written in Interval Notation: [ 0, 2 ] This function is increasing over the interval: 2 x < Written in Interval Notation: [ 2, ) Algebra I Page 19 Unit 2A. Graphs of Quadratic Functions
Average Rate of Change The Average Rate of Change on a Quadratic Function is the slope of the line segment between two points on the parabola. Example: 2 Find the average rate of change for the interval [0, 2]. The interval notation only tells us the x values. For the x value of 0, the y value is 1. The point is (0, 1) For the x value of 2, the y value is -3. The point is (2, -3) Find the slope of the line segment between the points (0, 1) and (2, -3). Slope Formula: m = y 2 y 1 x 2 x 1 m = 3 1 2 0 = 4 2 = 2 It makes sense that the average rate of change for this interval is negative because we previously stated that this was a decreasing interval. Algebra I Page 20 Unit 2A. Graphs of Quadratic Functions
Standard Form Quadratic Functions can be written in Standard Form: f (x) = ax 2 + bx + c Example: f (x) = 1x 2 3x 10 Functions written in Standard Form provide us hints about the graph of the function. The a value in f (x) = ax 2 + bx + c indicates whether the parabola opens up or down. o When a is positive, the parabola opens up and will have a minimum vertex. o When a is negative, the parabola opens down and will have a maximum vertex. The c value in f (x) = ax 2 + bx + c indicates the y-intercept of the parabola. This should look familiar to you. Remember when we studied linear functions, the constant term b in y = mx + b gave us the y-intercept of the line. Factored Form Quadratic Functions can be written in Factored Form: f (x) = a (x r1) (x r2) Example: f (x) = 1 (x 5) (x + 2) Functions written in Factored Form provide us hints about the graph of the function. The a value in f (x) = a (x r1) (x r2) indicates whether the parabola opens up or down. o When a is positive, the parabola opens up and will have a minimum vertex. o When a is negative, the parabola opens down and will have a maximum vertex. The r 1 and r 2 values in f (x) = a (x r 1 ) (x r 2 ) give us the zeros of the function. In the example: f (x) = 1 (x 5) (x + 2) the zeros are (5, 0) and (-2, 0) In the example: f (x) = -3 (x + 7) (x 6) the zeros are (-7, 0) and (6, 0) Did you notice that the x values for the points appears to be the opposite of the value shown in the equation? This happens because the formula for the equation uses subtraction. Algebra I Page 21 Unit 2A. Graphs of Quadratic Functions
Vertex Form Quadratic Functions can be written in Vertex Form: f (x) = a(x h) 2 + k Functions written in Vertex Form provide us hints about the graph of the function. The a value in f (x) = a (x h) 2 + k indicates whether the parabola opens up or down. o When a is positive, the parabola opens up and will have a minimum vertex. o When a is negative, the parabola opens down and will have a maximum vertex. The h value in f (x) = a(x h) 2 + k represents the x value of the vertex The k value in f (x) = a(x h ) 2 + k represents the y value of the vertex Example: f (x) = -2 (x 5) 2 + 3 The vertex is (5, 3) Example: f (x) = 1 (x + 7) 2 + 8 The vertex is (- 7, 8) Example: f (x) = 9 (x + 4) 2 6 The vertex is (- 4, -6) Notice that the x value appears to be the opposite of the number in the parentheses, while the y value is exactly the same as the number at the end of the equation. This is because the vertex form of the equation has a subtraction inside the parentheses. Algebra I Page 22 Unit 2A. Graphs of Quadratic Functions
Transformations There are four types of Transformations 1. Translation (slide) 2. Reflection (flip) (Source: http://www.regentsprep.org/regents /math/geometry/gt2/trans.htm) (Source: http://www.regentsprep.org/regents /math/geometry/gt1/reflect.htm) 3. Dilations (stretch or shrink) 4. Rotation (turn) This transformation will not be addressed with quadratic functions (Source: http://www.regentsprep.org/regents /math/geometry/gt3/ldilate2.htm) Algebra I Page 23 Unit 2A. Graphs of Quadratic Functions
Vertical Translation y = x 2 y = x 2 We can document the movement of the second parabola compared to the first using coordinate notation. (x, y) (x, y+1) Notice the 1 was added to the y value because the parabola shifted up one unit. It is easiest to see the shift at the vertex, but in fact ALL the points have shift up one unit. y = x 2 y = x 2 1 Coordinate Notation: (x, y) (x, y 1) Notice the 1 was subtracted from the y value because the parabola shifted down one unit. Algebra I Page 24 Unit 2A. Graphs of Quadratic Functions
Horizontal Translation y = x 2 y = (x ) 2 Coordinate Notation: (x, y) (x 1, y) Notice the 1 was subtracted from the x value because the parabola shifted left one unit. y = x 2 y = (x 1) 2 Coordinate Notation: (x, y) (x, y) Notice the 1 was added to the x value because the parabola shifted right one unit. Algebra I Page 25 Unit 2A. Graphs of Quadratic Functions
Vertical Reflection y = x 2 y = x 2 Coordinate Notation: (x, y) (x, y) Notice 1 was multiplied by the y value because the parabola flipped over the x-axis. Now all the y values are the opposite of what they had been. Remember x 2 means that you square the number and then take the opposite of it. Horizontal Reflection y = x 2 y = ( x) 2 Coordinate Notation: (x, y) ( x, y) Notice 1 was multiplied by the x value because the parabola flipped over the y-axis. Now all the x values are the opposite of what they had been. So why do the two graphs look the same? Think. If you take a number and square it, whether it starts out as a positive or negative number, it will always be positive after you square it, so the y values are exactly the same even if you use the opposite x-value! Those parentheses make all the difference! Algebra I Page 26 Unit 2A. Graphs of Quadratic Functions
Dilations: With a > 1 y = x 2 y = 4x 2 Coordinate Notation: (x, y) (x, 4y) The parabola was stretched by a factor of 4. Notice the point when the x-value is 1. The second graph shows that point four times higher than it was on the first graph. Yes, I know it looks like the parabola got smaller. But think of it this way. Imagine taking a Stretch Armstrong toy and stretching it. (He was popular back in the 1970 s) You would grab his arms and legs and pull. He looks thinner than he did before. Now imagine holding the ends of the parabola and stretching it. It looks thinner than it did before. Algebra I Page 27 Unit 2A. Graphs of Quadratic Functions
Dilations: With 0 < a < 1 y = x 2 y = 1 4 x2 Coordinate Notation: (x, y) (x, 1 4 y) The parabola was shrunk by a factor of 1 4. Notice the point when the x-value is 2. The second graph shows that point four times lower than it was on the first graph. Let s look at Stretch Armstrong again. Suppose you squish him vertically. He appears to get wider. The same thing happens to the parabola. Imagine pushing down on the parabola. It will spread out wider. Algebra I Page 28 Unit 2A. Graphs of Quadratic Functions
Transformations: Putting it all together So what happens when a parabola is transformed horizontally, vertically, reflected and dilated all at the same time? Let s start with a horizontal and vertical move at the same time. y = (x ) 2 3 y = x 2 Notice that the inside the parentheses moved the vertex of the parabola left 1 unit and the 3 outside the parentheses moved the vertex of the parabola down 3 units. Coordinate Notation: (x, y) (x 1, y 3) Algebra I Page 29 Unit 2A. Graphs of Quadratic Functions
Now let s throw in a reflection over the x-axis. y = (x ) 2 3 y = x 2 The vertex is still in the same place was after our last move, but notice how the negative in front of the parentheses has flipped the parabola over the x-axis. Coordinate Notation: (x, y) (x 1, y 3) Algebra I Page 30 Unit 2A. Graphs of Quadratic Functions
Finally let s stretch the parabola by a factor of 4 y = 4 (x ) 2 3 y = x 2 Now notice how the 4 in front of the parentheses has stretched the parabola. Coordinate Notation: (x, y) (x 1, 4y 3) Algebra I Page 31 Unit 2A. Graphs of Quadratic Functions