Transcript Ch. 9.5 power point

Chapter 9
Section 5
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More on Graphing Quadratic Equations:
Quadratic Functions
Graph quadratic equations of the form
y = ax2 + bx + c (a  0).
Use a graph to determine the number of
real solutions of a quadratic equation.
Use a quadratic function to solve an
application.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More on Graphing Quadratic Equations;
Quadratic Functions
In Section 5.4, we graphed the quadratic equation y = x2. By
plotting points, we obtained the graph of a parabola shown here.
Recall the lowest (or highest point if the parabola opens
downward) point on the graph is called the vertex of the
parabola. The vertical line through the vertex is called the axis,
or axis of symmetry. The two halves of the parabola are mirror
images of each other across this axis.
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Objective 1
Graph quadratic equations of the
form y = ax2 + bx + c (a  0).
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Graph quadratic equations of the form
y = ax2 + bx + c (a  0).
Every equation of the form
y = ax2 + bx + c
with a  0, has a graph that is a parabola. The vertex is
the most important point to locate when graphing a
quadratic equation.
Earlier in the course we solved linear equations in one variable that were
of the form Ax + B = C; then graphed linear equations in two variables
that were of the form y = mx + b. Now, we are ready to do the same sort
of thing for quadratic equations. We know how to solve ax2 + bx + c = 0;
and now we graph y = ax2 + bx + c.
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Graph quadratic equations of the form
y = ax2 + bx + c (a  0). (cont’d)
A procedure for graphing quadratic equations follows:
b

Step 1: Find the vertex. Let x = 2a , and find the
corresponding y-value by substituting for
x in the equation.
Step 2: Find the y-intercept.
Step 3: Find the x-intercepts (if they exist).
Step 4: Plot the intercepts and the vertex.
Step 5: Find and plot additional ordered pairs
near the vertex and intercepts as needed,
using symmetry about the axis of the
parabola.
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EXAMPLE 1
Graphing a Parabola by Finding
the Vertex and Intercepts
Graph y = x2 + 2x – 8.
Solution:
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EXAMPLE 2
Graphing a Parabola
Graph y = –x2 + 2x + 4.
Solution:
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Objective 2
Use a graph to determine the
number of real solutions of a
quadratic equation.
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Use a graph to determine the number of real
solutions of a quadratic equation.
Using the vertical line test from Section 3.6, we see that the
graph of an equation of the form
y = ax2 + bx + c
is the graph of a function. A function defined by an equation of
the form
(x) = ax2 + bx + c (a  0)
is called a quadratic function. The domain (possible x-values)
of a quadratic function is the set of all real numbers, or (–,);
the range (the resulting y-values) can be determined after the
function is graphed.
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EXAMPLE 3
Determining the Number of Real
Solutions from Graphs
Decide from the graph how many real number
solutions the corresponding equation x2 − 4 = 0 has.
Give the solution set.
Solution:
There are two real solutions,
{±2}, that correspond to the
graph.
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Objective 3
Use a quadratic function to solve
an application.
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EXAMPLE 4
Finding the Equation of a
Parabolic Satellite Dish
Suppose that a radio telescope has a parabolic dish
shape with diameter 350 ft and depth 48 ft. Find the
equation of the graph of a cross section.
Solution:
Let y = 48 ft, and x = 175 ft, solve for a.
y  ax2
48  a 175 
2
48  30,625a
48
a
30, 625
48
y
x2
30, 625
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