Aim #60: How do we graph a quadratic function from factored form,
f(x) = a(x - m)(x - n)?
Homework: Handout
Do Now: Solve the following equations by factoring:
2
2
2
1) x + 6x - 40 = 0
2) 2x + 11x = x - x - 32
2
1) Consider the equation y = x + 6x - 40
a. Given this quadratic equation, at what point(s) does the graph of this quadratic
equation cross the x-axis? Hint: What is the y-value when a graph crosses the
x-axis?
The ordered pairs above are called the x-intercepts of the graph.
The x-values alone, when the equation is equal to zero are called the roots
or zeros of the equation. The roots are the solution(s) where y = 0 and
these solutions correspond to the points where the graph of the equation
crosses the x-axis.
2
b. -Using our ROOTS, find the vertex for the graph y = x + 6x - 40.
-What is another method to find the vertex?
c. How could we find the y-intercept algebraically (where the graph crosses the
y-axis)?
d. Plot the points we have found so far.
Are there any other points we can determine?
2) How can we write a corresponding quadratic equation if we are given a pair of
roots? What if our roots are x = 3 or -6?
The factored form of a quadratic equation is: y = a(x - m)(x - n) where m, n are
the roots of the quadratic.
3) If we are given the roots and write our equation in standard or factored form,
what piece of information are we assuming?
4) For the following functions, identify the key features: the x-intercepts, vertex,
and the y-intercept:
a. f(x) = -(x + 2)(x - 5)
2
b. g(x) = x - 5x - 24
5) Identify the key features and graph the following functions. Choose appropriate
scales for your axes:
a. f(x) = 5(x - 2)(x - 3)
2
b. f(x) = x + 8x - 20
2
c. f(x) = -6x + 42x - 60
Sum It Up!!
• When a quadratic function is in factored form, we can find its x-intercepts, yintercept, axis of symmetry, and vertex.
• For any quadratic equation, the roots are the solution(s) where y = 0, and these
solutions correspond to the points where the graph of the equation crosses the
x-axis.
• A quadratic equation can be written in the form y = a(x - m)(x - n), where m and
n are the roots of the quadratic. Since the x-value of the vertex is the average
of the x-values of the two roots, we can substitute that value back into the
equation to find the y-value of the vertex. If we set x = o, we can find the yintercept.