Presentation on Graphs of Quadratic Functions
By Tim Wagner & David Lind
Math 599
As stated in the article, “A Historical Perspective from a to c” that we were assigned to
read, the NCTM Standards have long recommended that students explore the effects of changing
the coefficients in the quadratic function f(x) = ax2 + bx +c as a way to practice making
conjectures and using inductive reasoning. The Geogebra software provides an excellent method
for studying these changes because the graphs of quadratic functions can be manipulated and
changed so easily by using sliders for the values of a, b, and c in the formulas. The changes in
each coefficient result in changes in the graph in a smooth continuous way that makes
exploration of their effects much simpler and more visually apparent. Students can also explore
the effect on the number and properties of the roots of quadratic functions quite efficiently by
using the Geogebra software.
This exploration also allows students to learn about transformations, that is, changes in a
graph that follow specific patterns. Different types of transformations include: translations,
reflections, rotations, and dilations. Many of these transformations can be addressed by the
effects of changes in the coefficients of the quadratic function.
Let’s start by exploring the effects of changes in the coefficient a in the quadratic
function. If we begin with a = 1 and then slide to a =2, we observe that the rate of increase in y
with respect to x is doubled. This causes the graph to increase faster vertically and gives the
graph the visual appearance of being stretched vertically. If we slide to a = 3, the rate of increase
is tripled from the graph where a = 1, and the rate of stretching is even more pronounced.
Conversely, if we slide to a = 0.5, the rate of increase of y with respect to x is only half of that
for the case where a = 1, and the graph appears wider rather than narrower as in the previous
cases. This type of change in the graph is a special form of dilation, a transformation in which
the size of a graph is increased or decreased. When introducing this term to the students, it may
be helpful to refer to the situation where a person’s eyes are dilated – that is, where the size of
the pupils becomes larger or smaller in response to the amount of light entering them.
Next, notice what happens when the coefficient a becomes either zero or negative. When
a = 0, the squared term disappears and the graph becomes a straight line, either horizontal or
diagonal. If we slide to a = 1 and then to a = -1, we note that the graph turns upside down, but
has the same shape. In fact, the graph when a = -1 is the reflection of the graph when a =1 about
some line in the plane. In any case, when a is negative, the graph opens down rather than up.
Having explored the effect of changing a in the quadratic equation, let’s turn our
attention to the effect of changing the coefficient c. As c increases the graph moves upward, and
as c decreases the graph moves downward. Thus changing c results in a vertical translation of
the graph.
Finally, let’s examine the effect of changes in the coefficient b when a and c are held
constant. Starting at b = 0, as b increases the vertex of the graph moves to the left and down. As
b decreases, the vertex moves to the right and down. The path taken by the vertex looks
suspiciously like a parabola opening down with its vertex at the vertex of the original quadratic
equation. Let’s test this by setting a and c to 1 and b to 0. The vertex is at (0,1), so let plot y = 1
– x2, and see if the vertex follows this path as b changes. This works and in fact the graph of y =
c – x2 works for any value of c as long as b begins at 0 and a is 1. If a is 2, the path of the vertex
as b changes is steeper just as the original parabola is. So, let’s try the equation y = c – ax2 for
the path of the vertex as b changes. This works as well. So we now have an equation to represent
the path traversed by the parabola’s vertex as b changes. At the end of this document is an
algebraic derivation of the equation above.
Turning to the effect of changes in the values of the coefficients a, b, and c in the general
form of the quadratic equation on the nature of the roots or zeros produced, we observe the
following patterns. Please note that although we are presenting an algebraic argument here, we
want the students to induce these relationships by using the sliders for a, b, and c. This is
reflected in the text boxes included in the actual Geogebra file. Recall that a quadratic equation
may only have zero, one, or two real roots based upon the value of the discriminant (b2 – 4ac).
So, there are three possible cases as follows:
1) Zero real roots: b2 – 4ac < 0 or b2 < 4ac, which means that a and c must both be positive
or both be negative so that their product is positive (since b2 is positive). Also, although
this is much less intuitive, −2√𝑎𝑐 < 𝑏 < 2√𝑎𝑐 .
2) One real root: b2 – 4ac = 0 or b2 = 4ac, which means that:
a. When b = 0, either a or c must be zero and the other can be any number;
b. When b ≠ 0, a and c are both positive or both negative, along with b = ±2√𝑎𝑐 .
3) Two real roots: b2 – 4ac > 0 or b2 > 4ac, which means that:
a. When a and c are both positive or both negative, then 𝑏 < −2√𝑎𝑐 or 𝑏 > 2√𝑎𝑐
b. When either a or c or both are zero, then b ≠ 0
c. When a and c have opposite signs, then b can be any number
State Standards Addressed
8.11.08 Recognize and describe the general shape and properties of functions from graphs, tables,
or equations (e.g., linear, absolute value, quadratic, exponential, logarithmic).
8.11.10 Interpret the role of the coefficients and constants on the graphs of linear and quadratic
functions, given a set of equations.
8.11.11 Analyze functions by investigating domain, range, rates of change, intercepts, and zeros.
8.11.12 Create and connect representations that are tabular, graphic, numeric, and symbolic from a
set of data.
Derivation of formula for path of the vertex for changing values of b
y = ax2 + bx + c
y = a(x2 + (b/a)x) + c
y = a(x2 + (b/a)x + (b2/4a2)) + (c – (b2/4a2))
y = a(x + (b/2a))2 + (c – (b2/4a2))
So, the vertex of the original parabola is:
V = (-b/2a, c – (b2/4a2))
And, the equation of the path taken by the vertices as ‘b’
changes is the set of all ordered pairs (x,y) which fit the
pattern above for the vertices.
Thus, x = -b/2a and y = c – (b2/4a2).
Furthermore, y = c – (b2/4a2) = c – (-b/2a)2 = c – x2.
However, this equation is produced only by using the
vertex and does not include the ‘stretching’ factor of ‘a’ in
the standard form of the equation for the original
parabola. Including this factor of ‘a’ in the squared term
gives us the final form of the equation for the path of the
vertices as ‘b’ changes, specifically:
y = c – ax2.