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4.3 Graph Quadratic Functions in Vertex or Intercept Forms. Standard Form of a Quadratic. ax 2 + bx + c = 0. Find the axis of symmetry, state the vertex, and how it is concaved. -3x – x 2 + 4 = 8. AOS = x = -1.5 Vertex = (-1.5, -1.75) Concaved Down. Review: Graph of Vertex Form. - PowerPoint PPT Presentation
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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