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6.6 Analyzing Graphs of 6.6 Analyzing Graphs of Quadratic FunctionsQuadratic Functions

Graphing the parabola Graphing the parabola yy = = f f ((xx) = ) = axax22

Consider the equation y = x2

0 1 4 1

(–1, 1) (0, 0) (1, 1) (2, 4)

y

x

4

(–2, 4)

Axis of symmetry: x = 0 ( y=x2 is symmetric with respect to the y-axis )

Vertex(0, 0)

When a > 0, the parabola opens upwards and is called concave up.

The vertex is called a minimum point.

x – 2 – 1 0 1 2 y

For the function equation y = x2 , what is a ?

a = 1 . What if a does not equal 1?Consider the equation y = – 4x2 .

0 – 4 – 4

(0, 0)

– 16 y

– 16

(–2, –16)

(–1, –4) (1, –4) (–4, –16) x

When a < 0, the parabola opens downward and is called concave down.

The vertex is a maximum point.

x – 2 – 1 0 1 2 y

What is a ?

a = – 4

Properties of the Parabola Properties of the Parabola f f ((xx) =) = ax ax22

The graph of f (x) = ax2 is a parabola with the vertex at the origin and the y axis as the line of symmetry.If a is positive, the parabola opens upward, if a is

negative, the parabola opens downward.If a is greater than 1 (a > 1), the parabola is

narrower then the parabola f (x) = x2.If a is between 0 and 1 (0 < a < 1), the parabola is wider than the parabola f (x) = x2.

Algebraic Approach: y = – 4x2 – 3

x – 2 – 1 0 1 2-4x2 -16 -4 0 -4 -16-4x2-3 -19 -7 -3 -7 -19

Numerical Approach:Graphical Approach:

Graphing the parabola Graphing the parabola yy = = f f ((xx) = ) = axax22 + + k k

Consider the equation y = – 4x2 – 3 . What is a ?

a = – 4

x

Vertex(0, -3)

y = – 4x2 – 3 .

x

Vertex(0, -3)

y = – 4x2

In general the graph of y = ax2 + k is the graph of y = ax2 shifted vertically k units. If k > 0, the graph is shifted up. If k < 0, the graph is shifted down. (P. 267)

The graph y = – 4x2 is shifted down 3 units.

a = – 4. What effect does the 3 have on the function?

y

x

y= – 4x2 y= – 4(x – 3)2

Consider the equation y = – 4(x – 3)2 . What is a ?

The axis of symmetry is x = 3.

x – 2 – 1 0 1 2 –4 x2 -16 -4 0 -4 -16-4 (x-3) 2 -100 -64 -36 -16 -4

Numerical Approach:

3-360

Axis of symmetry is shifted 3 units to the right and becomes x = 3

Vertex Form of a Quadratic FunctionVertex Form of a Quadratic Function

The quadratic function f(x) = a(x – h)2 + k, a 0The graph of f is a parabola .Axis is the vertical line x = h.Vertex is the point (h, k).If a > 0, the parabola opens upward.If a < 0, the parabola opens downward.

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6.6 Analyzing Graphs of Quadratic Functions 6.6 Analyzing Graphs of Quadratic Functions

A family of graphs – a group of graphs that displays one or more similar characteristics.– Parent graph – y = x2

Notice that adding a constant to x2 moves the graph up. Notice that subtracting a constant from x before squaring it

moves the graph to the right.

y = x2

y = x2 + 2 y = (x – 3)2

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Vertex FormVertex Form Each function we just looked at can be written in the form

(x – h)2 + k, where (h , k) is the vertex of the parabola, and x = h is its axis of symmetry.

(x – h)2 + k – vertex form

Equation Vertex Axis of Symmetry

y = x2 or y = (x – 0)2 + 0

(0 , 0) x = 0

y = x2 + 2 ory = (x – 0)2 + 2

(0 , 2) x = 0

y = (x – 3)2 or y = (x – 3)2 + 0

(3 , 0) x = 3

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Graph TransformationsGraph TransformationsAs the values of h and k change, the graph of

y = a(x – h)2 + k is the graph of y = x2 translated.| h | units left if h is negative, or |h| units right

if h is positive.| k | units up if k is positive, or | k | units

down if k is negative.

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Graph a Quadratic Function in Vertex FormGraph a Quadratic Function in Vertex FormAnalyze y = (x + 2)2 + 1. Then draw its graph. This function can be rewritten as y = [x – (-2)]2 + 1.

– Then, h = -2 and k = 1 The vertex is at (h , k) = (-2 , 1), the axis of symmetry is x = -2. The

graph is the same shape as the graph of y = x2, but is translate 2 units left and 1 unit up.

Now use the information to draw the graph. Step 1 Plot the vertex (-2 , 1) Step 2 Draw the axis of

symmetry, x = -2. Step 3 Find and plot two

points on one side of the axis symmetry, such as (-1, 2) and (0 , 5).

Step 4 Use symmetry to complete the graph.

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Graph TransformationsGraph Transformations How does the value of a in the general form

y = a(x – h)2 + k affect a parabola? Compare the graphs of the following functions to the parent function, y = x2.

a.

b.

c.

d.

22y x

21

2y x

22y x

21

2y x

y = x2

a b

dc

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Example: Write a quadraticExample: Write a quadraticfunction for a parabola with a vertex of function for a parabola with a vertex of (-2,1) that passes through the point (1,-1).(-2,1) that passes through the point (1,-1).Since you know the vertex, use vertex form!

y=a(x-h)2+kPlug the vertex in for (h,k) and the other point

in for (x,y). Then, solve for a.-1=a(1-(-2))2+1

-1=a(3)2+1-2=9a

a9

21)2(

9

2 2

xy

Now plug in a, h, & k!

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Standard 9 Write a quadratic function in vertex form

Vertex form- Is a way of writing a quadratic equation that facilitates finding the vertex.

y – k = a(x – h)2

The h and the k represent the coordinates of the vertex in the form V(h, k).The “a” if it is positive it will mean that our parabola opens upward and if negative it will open downward.A small value for a will mean that our parabola is wider and vice versa.

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Standard 9 Write a quadratic function in vertex form

Write y = x2 – 10x + 22 in vertex form. Then identify the vertex.

y = x2 – 10x + 22 Write original function.

y + ? = (x2 –10x + ? ) + 22 Prepare to complete the square.

y + 25 = (x2 – 10x + 25) + 22Add –102

2( ) = (–5)2= 25 to each side.

y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared.

y + 3 = (x – 5)2 Write in vertex form.

The vertex form of the function is y + 3 = (x – 5)2. The vertex is (5, –3).

ANSWER

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EXAMPLE 1 Write a quadratic function in vertex form

Write a quadratic function for the parabola shown.

SOLUTION

Use vertex form because the vertex is given.

y – k = a(x – h)2 Vertex form

y = a(x – 1)2 – 2 Substitute 1 for h and –2 for k.

Use the other given point, (3, 2), to find a.2 = a(3 – 1)2 – 2 Substitute 3 for x and 2 for y.

2 = 4a – 2 Simplify coefficient of a.

1 = a Solve for a.

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EXAMPLE 1 Write a quadratic function in vertex form

A quadratic function for the parabola is y = (x – 1)2 – 2.

ANSWER

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EXAMPLE 1 Graph a quadratic function in vertex form

Graph y – 5 = – (x + 2)2.14

SOLUTION

STEP 1 Identify the constants a = – , h = – 2, and k = 5.

Because a < 0, the parabola opens down.

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STEP 2 Plot the vertex (h, k) = (– 2, 5) and draw the axis of symmetry x = – 2.

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EXAMPLE 1 Graph a quadratic function in vertex form

STEP 3 Evaluate the function for two values of x.

x = 0: y = (0 + 2)2 + 5 = 414

–

x = 2: y = (2 + 2)2 + 5 = 114

–

Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry.

STEP 4 Draw a parabola through the plotted points.

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GUIDED PRACTICE for Examples 1 and 2

Graph the function. Label the vertex and axis of symmetry.

1. y = (x + 2)2 – 3 2. y = –(x + 1)2 + 5

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GUIDED PRACTICE for Examples 1 and 2

3. f(x) = (x – 3)2 – 412