8-2 Quadratic Functions in Vertex Form

8-2 Quadratic Functions in Vertex Form

HSF.IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF.BF.B.3​ Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them

Open Ended

~Do Now~

How does the graph of g compare to the graph of the parent function $$f\left(x\right)=x^2$$  ?

$$g\left(x\right)\ =\ \left(x\ -2\right)^2$$  

Example 3: Understand the Graph of f(x) = ​(x - h)² + k

A. What information do the values of h and k provide about the graph of f(x) = (x - h)² + k

On the next slide you will see a series of graphs, each graph has a different value of "h" and "k."

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Pay close attention to the values of both "h" and "k" in each graph.​

Vertex : (1,-3)​

​The "h" value is inside the parentheses and ALWAYS comes after a subtraction side.

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f(x) = (x - 1)²-3

Vertex: (1, 2)

f(x) = (x - 1)² + 2

Vertex: (-2, -1)

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If "h" comes after a plus sign, that means (x - -h)². So h is negative.​

f(x) = (x + 2)² -1

• The values of h and k determine the location of the vertex and axis of symmetry of the parabola.​

• The vertex of the graph of

  f(x) = (x - h)² + k​

  is at (h, k).

• The axis of symmetry is x = h​

Multiple Choice

What is the axis of symmetry of the following function;

f(x) = (x - 4)² - 8

• x = 4

• x = -4

• x = 8

• x = -8

Example 3: Understand the Graph of f(x) = a​(x - h)² + k

B. How does the value of a affect the graph of f(x) = a(x - h)² + k

On the next slide you will see a series of graphs, each graph has a different value of "a."

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Pay close attention to the values of "a" in each graph.​

f(x) = (x - 1)² - 3

g(x) = 2(x - 1)² - 3​

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f(x) = (x - 1)² + 2

g(x) = 0.25(x - 1)² + 2​

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f(x) = (x + 2)² - 1

g(x) = -0.1(x + 2)² - 1​

• The value of a does not affect the location of the vertex.

• The sign of a affects the direction of the parabola.

  • a > 0 it opens upwards

  • a < 0 it opens downwards​

• The absolute value of a affects the width of the parabola.

  • |a| > 1 the parabola is more narrow than the parent function.

  • 0 < |a| < 1 the parabola is wider than the parent function.

​The function f(x) = a(x - h)² + k, where a ≠ 0 is called vertex form of a quadratic function.

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• The vertex of the graph is (h, k)

• The graph of f(x) = a(x - h)² + k is a translation of the function f(x)=ax², that is translated h units horizontally and k units vertically.

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*** Remember, if you see a quadratic function in vertex form with a plus sign within the parentheses, that means it is a double negative. So h is negative!!!!!!!

Multiple Select

How does the graph of f(x) = -3(x - 5)² + 7 compare to the graph of the parent function? Select all that apply.

• It is translated 5 units to the right

• It is translated up 7 units

• It opens the same way.

• It is more narrow .

Example 4: Graph Using Vertex Form

Graph the function f(x) = -2(x + 1)² + 5 [ This is the same as f(x)=-2(x - (-1))² + 5 ]

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​Step 1: Plot the vertex and axis of symmetry.

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Step 2: Evaluate the function to find two other points.

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Step 3: Reflect the points across the axis of symmetry

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Step 4: Draw a parabola through the points.​

​Find the vertex and axis of symmetry, and sketch the graph of the function?

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1. g(x) = -3(x - 2)² + 1

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2. h(x) = (x + 1)² - 4​

Poll

How do you feel so far about this lesson?

• I am a PRO fo SHO

• I actually understand most of what is going on

• It makes some sense but I'm still confused

• The struggle is real

​Homework:

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If you did not complete the 2 graphs from slide 15, complete those for class on Monday.​

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Please make sure to complete "Topic 8: Math XL 8-2" ​