Intro to Quadratics Graphically from Lines
Consider the following:
y = 2x+8
y = -x + 2
or
f(x) = 2x+8
g(x) = -x+2
Graph these on your calculator and sketch
the picture at right.
Sum of two linear functions
Adding these functions together yield f(x) +
g(x) = 2x + 8 + -x + 2 = x + 10, which
we’ll call s(x)….for sum of the two
functions.
Notice the y value for f(x) at x = -1 is f(-1) = 6. g(-1) = 3. So, f(-1) + g(-1) = 9. This is
also found by plugging in x = - 1 into s(x). s(-1) = 9. We could do this for each of the
values to generate the new graph of the sum of f(x) and g(x).
Fill out the chart below:
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
f(x)
-4
-2
0
2
4
6
8
10
12
14
16
18
20
g(x)
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
s(x)=f(x)+g(x)
4
Plot these s(x) points on the graph above.
When you add these to linear functions,
the result is another linear function.
Compare the points plotted and the
equation for s(x).
9
Locate the x – intercepts of f(x) and g(x).
What happens on s(x)?
Look at the intersection of f(x) and g(x).
What happens on s(x)?
Mark Thomas
Stillwater High School
mthomas@stillwaterschools.com
Product of two linear functions
Now we wish to multiply f(x) and g(x). We will call it p(x) = f(x)*g(x) for product.
p(-3) = f(-3)*g(-3) = (2)(5) = 10. We could multiply through the chart again and plot the
points.
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
f(x)
-4
-2
0
2
4
6
8
10
12
14
16
18
20
g(x)
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
s(x)=f(x)+g(x)
4
5
6
7
8
9
10
11
12
13
14
15
16
p(x)=f(x)*g(x)
What happens to the function
values on p(x) compared to
s(x)?
10
Plot the points for p(x) for x values from -5
to 3.
This shape is called a parabola and is given
by a quadratic function.
What do you observe for the x-intercepts for
f(x) and g(x)?
What happens when f(x) = g(x)?
What are the coordinates for the highest
point (maximum) of the parabola? How does
the x-value compare to the x-intercepts of
f(x) and g(x)?
What must be true about f(x) and g(x) in order for p(x)….
to be positive? (above the x-axis)
to be negative? (below the x-axis)
to be zero? (on the x-axis)
to have a minimum point instead of a maximum point?
Mark Thomas
Stillwater High School
mthomas@stillwaterschools.com
Summary
Distributing (foiling) f(x)*g(x)
= (2x + 8)(-x + 2) = -2x2 + 4x – 8x + 16
= -2x2 – 4x + 16.
If y = ax2 + bx + c, then c = 16. What is
special about this on the graph?
The vertex is the maximum (or
minimum) of the parabola.
b
It is found by x 
. How do you find
2a
the y value?
The zeros (x-intercepts) of the parabola
are the x-intercepts of the linear pieces
that are multiplied to give the quadratic.
Extension
Draw two lines where the multiplication involves two negatives to get a positive product.
Draw two lines where the parabola opens upward.
What must be true about the lines in order for the parabola to open upward?
Mark Thomas
Stillwater High School
mthomas@stillwaterschools.com