10.4
Transcript of 10.4
10.4 Solve Quadratic Equations by Graphing
VocabularyQuadratic Equation – an equation that can be
written in the standard formwhere
Zero(s) of a function – x value(s) for which
* Zero(s) of a polynomial function and root(s) of a polynomial are the same!
Solve by Graphing (vs. Factoring)Recall Solve by Factoring:
Solve by Graphing:
Solve a quadratic equation having two solutionsExample 1
SOLUTION
STEP 1 Write the equation in standard form.
Solve by graphing.3=x2 2x–
Write original equation.3=x2 2x–
0=x2 2x– 3– Subtract 3 from each side.
STEP 2 Graph the related function . The x-intercepts are –1 and 3.y = x2 2x– 3–
Solve a quadratic equation having two solutionsExample 1
CHECK You can check –1 and 3 in the original equation.
Write original equation.3=x2 2x– 3=x2 2x–
3 3 Substitute for x.=?
=?( )21– 2– ( )1– – )32()23(
3=3 3=3 Simplify. Each solution checks.
ANSWER
The solutions of the equation are 1 and 3.3=x2 2x– –
Solve a quadratic equation having one solutionExample 2
Solve by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
1=– x2 2x+
Write original equation.1=– x2 2x+
Subtract 1 from each side.0=x2 2x+ 1––
STEP 2 Graph the related function . The x-intercept is 1.
y = – x2 2x+ 1–
Solve a quadratic equation having one solutionExample 2
The solution of the equation is 1.
ANSWER
1=– x2 2x+
Solve a quadratic equation having no solutionExample 3
Solve by graphing.
SOLUTION
STEP 1 Write the equation in standard form.
4x=x2 7+
Write original equation.4x=x2 7+
0=x2 4x– 7+ Subtract 4x from each side.
STEP 2 Graph the related function The graph has no x-intercepts.
y = x2 4x– 7.+
Solve a quadratic equation having no solutionExample 3
The equation has no solution.4x=x2 7+
ANSWER
Number of Solutions of a Quadratic EquationTwo Solutions
A quadratic equation has two solutions if the graph of its related function has two x-intercepts.
One Solution
A quadratic equation has one solution if the graph of its related function has one x-intercept.
No Solution
A quadratic equation has no real solution if the graph of its related function has no x-intercepts.
The graph of the equation
what value or values of x is y 0?
ANSWER The correct answer is C.
Multiple Choice PracticeExample 4
SOLUTION
You can see from the graph that the x-intercepts are 7 and 1. So, y 0 when x 7 and x 1.= =
–= –
y = 6x+x2 7 is shown. For –=
– x = 7 only x = 1 only
x = –7 and x = 1 x = –1 and x = 7
Approximate the zeros of a quadratic function
Approximate the zeros of to the nearest tenth.
Example 5
y = 4x+x2 1+
SOLUTION
STEP 1 Graph the function There aretwo x-intercepts: one between 4
another between 1 and 0.
y = 4x+x2 1.+
–– –and 3 and
Approximate the zeros of a quadratic functionExample 5
STEP 2 Make a table of values for x-values between 4 and 3 and between 1 and 0 using anincrement of 0.1. Look for a change in the signs of the function values.
– – –
x
y
– 3.9 – 3.8 – 3.7 – 3.6 – 3.5 – 3.4 – 3.3 – 3.2 – 3.1
– 0.11 – 0.44 – 0.75 – 1.04 – 1.31 – 1.56 – 1.790.240.61
x
y
– 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1
– 1.31 – 1.04 – 0.75 – 0.44 – 0.11 0.24 0.61– 1.56– 1.79
Approximate the zeros of a quadratic functionExample 5
In each table, the function value closest to 0 is – 0.11. So, the zeros of are about – 3.7 and about – 0.3.
y = 4x+x2 1+
ANSWER
Solve a multi-step problemExample 6
An athlete throws a shot put with an initial vertical velocity of 40 feet per second.
SPORTS
a. Write an equation that models the height h (in feet) of the shot put as a function of the time t (in seconds) after it is thrown.
b. Use the equation to find the time that the shot put is in the air.
Solve a multi-step problemExample 6
SOLUTION
a. Use the initial vertical velocity and height to write a vertical motion model.
16t2=h Vertical motion model+ vt + s–
Substitute 40 for v and 6.5 for s.16t2=h + 40t + 6.5–
b. The shot put lands when h 0. To find the time twhen h 0, solve for t.
== 16t
2=0 + 40t + 6.5–
To solve the equation, graph the related functionon a graphing calculator. Use the trace feature to find the t-intercepts.
16t2=h + 40t + 6.5–
Solve a multi-step problemExample 6
There is only one positive t-intercept. The shot put is in the air for about 2.6 seconds.
ANSWER
Relating Roots of Polynomials, Solutions of Equations, x-intercepts of Graphs, and Zeros of Functions
The Roots of the Polynomial are 2 & 6.
The Solutions of the equation are 2 & 6.
The x-intercepts of the graph of occur where y=0, so the x-intercepts are 2 and 6.
The Zeros of the function are the values of x for which y=0, so the zeros are 2 and 6.
10.4 Warm-UpSolve the equation by graphing.
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