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Students Will: Graph and transform quadratic functions. Quadratic Functions Part: Section Topics 7 & 8 Students Will: Graph and transform quadratic functions.

x = 0; (0, –4); opens downward Quadratic Functions Part: Section Topics 7 & 8 Let’s Recall For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y = x2 + 3 2. y = 2x2 3. y = –0.5x2 – 4 x = 0; (0, 3); opens upward x = 0; (0, 0); opens upward x = 0; (0, –4); opens downward

For the parent function f(x) = x2: Quadratic Functions Part: Section Topics 7 & 8 The quadratic parent function is f(x) = x2. The graph of all other quadratic functions are transformations of the graph of f(x) = x2. For the parent function f(x) = x2: The axis of symmetry is x = 0, or the y-axis. The vertex is (0, 0) The function has only one zero, 0.

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can move it up or down by adding a constant to the y-value: Note: to move the line down, we use a negative value for C. C > 0 moves it up C < 0 moves it down  

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can move it left or right by adding a constant to the x-value: Subtracting C moves the function to the right (the positive direction) C > 0 moves it up C < 0 moves it down  

add to y, go high add to x, go left Quadratic Functions Part: Section Topics 7 & 8 A way to remember what happens to the graph when we add a constant: add to y, go high add to x, go left

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can stretch or compress it in the y-direction by multiplying the whole function bya constant C > 1 stretches it 0 < C < 1 compresses

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can stretch or compress it in the x-direction by multiplying x by a constant. C > 1 compresses it 0 < C < 1 stretches it

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can stretch or compress it in the y-direction by multiplying the whole function bya constant C > 1 stretches it 0 < C < 1 compresses

Quadratic Functions Part: Section Topics 7 & 8  This is the quadratic parent function f(x) = x2 We can flip it upside down by multiplying the whole function by −1: This is also called:  reflection about the x-axis We can combine a negative value with a scaling:

Summary Quadratic Functions Part: Section Topics 7 & 8 y = f(x) + C C > 0 moves it up C < 0 moves it down y = f(x + C) C > 0 moves it left C < 0 moves it right y = Cf(x) C > 1 stretches it in the y-direction 0 < C < 1 compresses it y = f(Cx) C > 1 compresses it in the x-direction 0 < C < 1 stretches it y = −f(x) Reflects it about x-axis y = f(−x) Reflects it about y-axis

Summary with Examples f(x) = x2 + 2 f(x) = x2 − 3 f(x) = (x−4)2 Quadratic Functions Part: Section Topics 7 & 8 Summary with Examples Move 2 spaces up:   f(x) = x2 + 2 Move 3 spaces down: f(x) = x2 − 3 Move 4 spaces to the right: f(x) = (x−4)2  Move 5 spaces to the left: f(x) = (x+5)2   Stretch it by 2 in the y-direction: f(x) = 2x2  Compress it by 3 in the x-direction: f(x) = (3x)2  Flip it upside down: f(x) = −x2 

Quadratic Functions Part: Section Topics 7 & 8

Quadratic Functions Part: Section Topics 7 & 8 The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

Order the functions from narrowest graph to widest. Quadratic Functions Part: Section Topics 7 & 8 Order the functions from narrowest graph to widest. f(x) = 3x2, g(x) = 0.5x2 Step 1 Find |a| for each function. |3| = 3 |0.05| = 0.05 Step 2 Order the functions. f(x) = 3x2 g(x) = 0.5x2 The function with the narrowest graph has the greatest |a|.

Example 1B: Comparing Widths of Parabolas Quadratic Functions Part: Section Topics 7 & 8 Example 1B: Comparing Widths of Parabolas Order the functions from narrowest graph to widest. f(x) = x2, g(x) = x2, h(x) = –2x2 Step 1 Find |a| for each function. |1| = 1 |–2| = 2 Step 2 Order the functions. h(x) = –2x2 The function with the narrowest graph has the greatest |a|. f(x) = x2 g(x) = x2

Quadratic Functions Part: Section Topics 7 & 8

Quadratic Functions Part: Section Topics 7 & 8 The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of f(x) = ax2 up or down the y-axis.

Quadratic Functions Part: Section Topics 7 & 8

Compare the graph of the function with the graph of f(x) = x2 Quadratic Functions Part: Section Topics 7 & 8 Example 2A Continued Compare the graph of the function with the graph of f(x) = x2 g(x) = x2 + 3 The vertex of f(x) = x2 is (0, 0). g(x) = x2 + 3 is translated 3 units up to (0, 3). The axis of symmetry is the same.

Compare the graph of each the graph of f(x) = x2. Quadratic Functions Part: Section Topics 7 & 8 Compare the graph of each the graph of f(x) = x2. g(x) = –x2 – 4 Method 1 Compare the graphs. The graph of g(x) = –x2 – 4 opens downward and the graph of f(x) = x2 opens upward. The axis of symmetry is the same. The vertex of g(x) = –x2 – 4 f(x) = x2 is (0, 0). is translated 4 units down to (0, –4). The vertex of

Compare the graph of the function with the graph of f(x) = x2. Quadratic Functions Part: Section Topics 7 & 8 Compare the graph of the function with the graph of f(x) = x2. g(x) = x2 + 2 Method 1 Compare the graphs. The graph of g(x) = x2 + 2 is wider than the graph of f(x) = x2. The graph of g(x) = x2 + 2 opens upward and the graph of f(x) = x2 opens upward.

Graphing Vertex Form:

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