a Substitute the given values to write the function. To graph it, begin by finding its axis of symmetry and make table of values using x values around the axis of symmetry.
B
b Use the Zero Product Property.
C
c Proceed in the same way as you did in Part A to graph the functions.
D
d Consider the zeros, axes of symmetry and leading coefficients of the graphs.
E
e Consider the factored form of a function. What are the solutions of the function?
A
aFunction: f(x)=x^2+x-6
Graph:
B
b x=2 and x=-3
C
cGraph:
D
d See solution.
E
e See solution.
Practice makes perfect
a Let's begin by substituting a= 1, p=2, and q=-3 in the equation. Then we can simplify it.
Notice that the leading coefficient is positive. Therefore, the graph opens upward and it has a minimum value on its vertex. To graph the equation, we will first find its axis of symmetry by using the formula x=- b2 a. Let's first highlight the coefficient of the function and determine a and b.
f(x)= 1x^2+ 1x -6
Therefore, a= 1 and b= 1. Now, we can substitute these values into the formula, so we can find the axis of symmetry.
Therefore, the solutions of the equation are 2 and -3.
c We will begin by determining the functions for a=4, -3 and 12.
a
a(x-2)(x+3)
f(x)
4
4(x-2)(x+3)
4x^2+4x-24
-3
-3(x-2)(x+3)
-3x^2-3x+18
1/2
1/2(x-2)(x+3)
1/2x^2+1/2x-3
To graph the functions, we will proceed in the same way as we did in Part A. Let's first find their axes of symmetry.
f(x)
x=-b/2a
Axis of Symmetry
4x^2+ 4x -24
x=-4/2( 4)
x=-1/2
-3x^2 -3x+ 18
x=--3/2( -3)
x=-1/2
1/2x^2+ 1/2x -3
x=-1/2/2( 1/2)
x=-1/2
Now, we can graph these functions as we did in Part A. Notice that the leading coefficient of the second function is negative, so the graph of it opens downward. The other two functions open upward because of the positive leading coefficient.
d By looking at the graphs in Part C, we can list the similarities and differences as shown below.
All three curves have the same zeros and the same axes of symmetry.
The graphs with a positive leading coefficients open upward and the graph with negative leading coefficient opens downward.
The graphs with leading coefficients a=4 and a=-3 are much steeper than the graph with a leading coefficient a= 12.
e Let's consider the factored form of a quadratic function.
f(x)=a(x-p)(x-q)
When a quadratic function is in factored form as shown, we can immediately determine its solutions as p and q by the Zero Product Property.
