4.3 graph quadratic functions in vertex or intercept forms

4.3 Graph Quadratic Functions in Vertex or

Intercept Forms

Standard Form of a Quadratic

ax2 + bx + c = 0Find the axis of symmetry, state the vertex,

and how it is concaved.-3x – x2 + 4 = 8

AOS = x = -1.5Vertex = (-1.5, -1.75)Concaved Down

Review: Graph of Vertex Form

Example 1: Graph a Quadratic Function in

Vertex FormGraph 5)2(

41 2 xy

Step 1: Identify the constants (a, h, and k) a = -1/4 h = -2 k = 5

Step 2: Plot the vertex and draw the axis of symmetry

Remember that (h, k) is your vertex!!

Example 1: Graph a Quadratic Function in

Vertex FormGraph 5)2(

41 2 xy

Step 3: Evaluate the function for one value of x. Choose an easy one!!

For example, use x = 0!y = -1/4(x + 2) + 5 2

y = -1/4(0 + 2) + 5 2

y = -1/4(2) + 5 2

y = -1/4(4) + 5y = -1+ 5y = 4 Plot (0,

4)

Example 1: Graph a Quadratic Function in Vertex

FormGraph 5)2(

41 2 xy

Step 4: Reflect the point over the axis of symmetry.

What do you get?(-4, 4)

Step 5: Draw a parabola through your points!

Your Turn!

3)2( 2 xy

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

Vertex: (-2, -3) AOS: x = -2

Use a Quadratic Model in Vertex Form

The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function

where x and y are measured in feet. What is the distance d between the two towers?

27)1400(70001 2 xy

Use a Quadratic Model in Vertex Form

SOLUTIONWhat is the vertex?(1400, 27)

What does that mean?Remember that the vertex is in the middle of the parabola That means that each tower is 1400 feet from the midpoint.So…? The distance between the two towers is 2(1400) or 2800ft.

Graph of Factored Form y = a(x – p)(x – q)

Example 3: Graph a Quadratic Function in

factored Form)1)(3(2 xxyGraph

Step 1: Identify the x-intercepts. Remember it’s the opposite of what you think!! If (x + 3), then p = ______

If (x – 1), then q = ______-31

Plot the intercepts!!!

Example 3: Graph a Quadratic Function in

factored Form

2qpx

)1)(3(2 xxy

213

Graph Step 2: Find the x-coordinate of the vertex.

= -1Step 3: Find the y-coordinate of the vertex.y = 2(x + 3)(x – 1)

y = 2(-1 + 3)(-1 – 1)y = 2(2)(-2)y = -8

So…plot the vertex (-1, -8) and connect the points!

Example 4: Change from Factored to Standard

FormWrite y = -2(x + 5)(x – 8) in standard form.

1. Write original function

y = -2(x + 5)(x – 8)

2. Multiply using FOIL orDouble Distribution3. Combine like terms

4. Distribute the -2 to get standard form

y = -2(x - 8x + 5x – 40) 2

y = -2(x - 3x – 40) 2

y = -2x + 6x + 802

Example 5: Change from Vertex

to Standard Form

1. Write original function

3. Multiply using FOIL orDouble Distribution

Write f(x) = 4(x - 1) +9 in standard form.2

f(x) = 4(x – 1) + 9

2

f(x) = 4(x – 1)(x – 1) + 92. Rewrite (x – 1) 2

f(x) = 4(x - 1x – 1x + 1) + 9 2

Example 5: Change from Vertex

to Standard Form

4. Combine like terms

5. Distribute the 4

Write f(x) = 4(x - 1) +9 in standard form.2

f(x) = 4(x - 2x + 1) + 9 2

f(x) = (4x - 8x + 4) + 9 2

6. Combine like terms to get standard form

f(x) = 4x - 8x + 13 2

Re-write the equation in Standard Form:

y = 3(x – 1)(x + 2)

Re-write the equation in Standard Form:

y = -1(x + 3)2 – 4