Download Notes 9-2: Solving Quadratic Equations by Graphing

Notes 9-2: Solving Quadratic
Equations by Graphing
I. Solutions vs Roots vs Zeros vs X-intercepts
Just as we can solve a quadratic equation by Zero Product
Property, Square Root Property, Completing the Square, or
the Quadratic Formula, we can also solve by graphing. We
graph the related function and look for the x-intercepts.
The x-intercepts of a quadratic function show the solutions of a
quadratic equation.
The x-coordinate of the x-intercept is called a zero of the function.
The x-intercepts of a quadratic function written in the form y = (x p)(x - q) are (p, 0) and (q, 0).
Remember that quadratic equations can have two solutions, one
solution, or zero real solution (two imaginary solutions). This
means that the related functions can have two x-intercepts, one xintercept, or no x-intercept (we cannot graph imaginary numbers
on the Cartesian plane.
Quadratic Equation
0 = ax2 + bx + c
0 = x2 + 4x + 3
Solutions: x = -1 or x = -3
Roots: -1 or -3
0 = -x2 + 6x - 9
Solution: x = 3
Roots: 3
0 = x2 + 4
Solutions: no real solutions,
two imaginary solutions
Roots: no real roots; two
imaginary roots
Graph
Quadratic Function
y = ax2 + bx + c
f(x) = x2 + 4x + 3
X-intercepts: (-1, 0) and (-3,
0)
Zeros: -1 and -3
f(x) = -x2 + 6x - 9
X-intercepts (3, 0)
Zeros: 3
f(x) = x2 + 4
X-intercepts: none
Zeros: nonte
One equation, many different types of problems
x2 + x – 6 = 0
“Solve x2 + x – 6 = 0” or
“Find the solutions to x2 + x – 6 = 0”
(x – 2) (x + 3) = 0
(x – 2) = 0
(x + 3) = 0
The solutions are x = 2 and x = -3
Graph x2 + x – 6 = 0
The x-intercepts are 2 and -3.
The zeros are 2 and -3.
“Find the roots of x2 + x – 6 = 0
(x – 2) (x + 3) = 0
(x – 2) = 0
(x + 3) = 0
The roots are 2 and -3
II. Approximating Zeros
• Solving by graphing is great when you have an integer solution.
However, sometimes you will have to approximate your answer.
For example, find the approximate zeros of the quadratic function
y = x2 + 2x – 4.
The zeros are approximately -3.42
and 1.24. How could we get exact
solutions?
Use quadratic formula and keep it in
radical form.
III. Real-World Applications
High Dive The height of a diver's jump can be
represented by a quadratic function. The standard
height in a platform high dive competition is 10
meters. If diver pushes off the platform with a
velocity of 8 meters per second the function that
models the diver’s height y after x seconds is y = –
4.9x2 +8x +10.
http://www.nytimes.com/interactive/2012/08/09/spo
rts/olympics/diving-how-to-win.html?_r=0