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Graphing Quadratic Functions

Warm-up
1. Evaluate the expression for b =- 3 and a = 2.

2. Find the value of in the equation when b = 4 and a = -2.

3. Find the value of y in the equation when x = 0.

4. Find the value of y in the equation when .

5. Find the approximate value of y to two decimal places in the equation
when x =1.5.

Answers to warm-up

1. –3
2. x = 1
3. y = 5
4. y = -3/4
5. y≈ 3.47

Today we will:

1. Understand how the coefficients of a quadratic function influence its graph

a. The direction it opens
b. Its vertex
c. Its line of symmetry
d. Its y-intercepts

Tomorrow we will:

1. Explore translations of parabolas
.
Parabolas

The path of a jump shot as the ball travels toward the basket is a parabola.

Key terms

Parabola – a curve that can be modeled with a quadratic function.
Quadratic function – a function that can be written in the form
y = ax2 + bx + c, where a ≠ 0.
Standard form of a quadratic function – the form y = ax2 + bx + c, where a ≠ 0.
Vertex – the point where a parabola crosses its line of symmetry.
Maximum – the vertex of a parabola that opens downward. The y- coordinate of the
vertex
is the maximum value of the function.
Minimum – the vertex of a parabola that opens upward. The y-coordinate of the
vertex is the minimum value of the function.
y-intercept – the y-coordinate of the point where a graph crosses the y-axis.
x-intercept – the x-coordinate of the point where a graph crosses the x-axis.

The graph of the quadratic function y = ax2 + bx + c, where a ≠ 0, is a parabola.
If a is positive
the graph opens up
the vertex is a minimum
If a is negative
the graph opens down
the vertex is a maximum

The line of symmetry is the vertical line . The x-coordinate of the vertex is .
To find the y-coordinate of the vertex, substitute   for x in the function and solve for y .
The y- intercept of the graph of a quadratic function is c.

Example 1
Choose the function that models the
parabola at the right.
A. y = −0.5x2 + 4x + 5
B. y = 0.5x2 + 4x − 3
C. y = −0.5x2 + 4x − 3
D. y = −0.4x2 + 4x − 3
E. y = x2 + 4x + 5

Solution
The graph opens down so a is negative.
B and E are out.
The y-intercept is –3.
A is out.
Find the line of symmetry.

Choice C:

Choice D:

The line of symmetry is x = 4.
C is the correct function.

Example 2
Use the function y = 2x2 + 3x −1
A. Tell whether the graph opens up or down.
B. Tell whether the vertex is a maximum or a minimum.
C. Find an equation for the line of symmetry.
D. Find the coordinates of the vertex.

Solution
A. a is positive, so the graph opens up.
B. The vertex is a minimum.
Parabola

 

Example 3
Use the quadratic function y = 3x2 −18x + 25
A. Without graphing, will the graph open up or down?
B. Is the vertex a minimum or a maximum?
C. What is the equation of the line of symmetry?
D. Find the coordinates of the vertex of the graph.
E. Find the y-intercept.
F. Graph the function.

Solution

A. a is positive, so the graph will open up.
B. The vertex is a minimum

F.

Example 4
Use the function y = x2 + 0.6x − 7.75
A. Find the y-intercept of the graph.
B. Use a graph to estimate the x-intercepts. Check one x- intercept by substitution .

Solution
A. The y-intercept is c or –7.75
B. The x-intercepts are 2.5 and –3.1

Check: Substitute 2.5 for x in the original equation.

Example 5
Match each equation with its graph.

Solution:
A. 4
B. 3
C. 2
D. 1

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