10.1: Quadratic Functions and Their Graphs
Suppose
Graphs of Quadratic Functions
Previous Concepts introduced the concept of factoring quadratic trinomials of the form
Example A
Graph the most basic quadratic equation,



–2  4 
–1  1 
0  0 
1  1 
2  4 
Solution:
By graphing the points in the table, you can see that the shape is approximately like the graph below. This shape is called a parabola.
The Anatomy of a Parabola
A parabola can be divided in half by a vertical line. Because of this, parabolas have symmetry. The vertical line dividing the parabola into two equal portions is called the line of symmetry. All parabolas have a vertex, the ordered pair that represents the bottom (or the top) of the curve.
The vertex of a parabola is the ordered pair
Because the line of symmetry is a vertical line, its equation has the form
As with linear equations, the
An equation of the form
If
If
The variable
If
If
Example B
Find the
Solution:
To find the
This means that
Thus the
Finding the Vertex of a Quadratic Equation in Standard Form
The
Example C
Determine the direction, shape and vertex of the parabola formed by
Solution:
The value of
 Because
a is negative, the parabola opens downward.  Because
a is between –1 and 1, the parabola is wide about its line of symmetry.  Because there is no
b term,b=0 . Substituting this into the equation for thex coordinate of the vertex,x=−b2a=−02a=0 . (Note: It does not matter whata equals; sinceb=0 , the fraction equals zero.) To find they coordinate, substitute thex coordinate into the equation:
The vertex is
Domain and Range
Several times throughout this textbook, you have experienced the terms domain and range. Remember:
 Domain is the set of all inputs (
x− coordinates).  Range is the set of all outputs (\begin{align*}y\end{align*}
y− coordinates).
The domain of every quadratic equation is all real numbers \begin{align*}(\mathbb{R})\end{align*}. The range of a parabola depends upon whether the parabola opens up or down.
If \begin{align*}a\end{align*} is positive, the range will be \begin{align*}y \ge k\end{align*}.
If \begin{align*}a\end{align*} is negative, the range will be \begin{align*}y \le k\end{align*}, where \begin{align*}k= \text{the }y\end{align*}coordinate of the vertex.
Example D
Find the range of the quadratic function \begin{align*}y=2x^2+16x+5\end{align*}.
Solution:
To find the range, we must find the \begin{align*}y\end{align*}value of the vertex. Using the formula given above, we can find the \begin{align*}x\end{align*}value of the vertex, and use that to find the \begin{align*}y\end{align*}value of the vertex.
Since the \begin{align*}x\end{align*}value of the vertex is \begin{align*}x=\frac{b}{2a}\end{align*}, we get \begin{align*}x=\frac{16}{2\cdot 2}=4.\end{align*}
Now, substitute 4 into the function:
\begin{align*}y=2x^2+6x+5=2(4)^2+6(4)+5=2(16)+24+5=32+29=3\end{align*}
The \begin{align*}y\end{align*}value of the vertex is 3, and since \begin{align*}a=2\end{align*}, the parabola is facing down, so 3 is the highest possible value for the range.
Thus, the range is \begin{align*}y\le 3\end{align*}.
Guided Practice
Determine the direction, vertex and range of \begin{align*}y=7x^2+14x9\end{align*}.
Solution:
Since \begin{align*}a=7\end{align*} is positive, the direction of the parabola is upward. Now we find the vertex:
\begin{align*}x=\frac{b}{2a}=\frac{14}{2\cdot 7}=\frac{14}{14}=1.\end{align*}
Now, substitute \begin{align*}x=1\end{align*} into the quadratic function:
\begin{align*}y=7x^2+14x9=7(1)^2+14(1)9=7(1)149=723=16.\end{align*}
Thus, the vertex is (1, 16).
Since the parabola faces up, and the \begin{align*}y\end{align*}value of the vertex is 16, the range is \begin{align*}y\ge 16\end{align*}.
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Graphs of Quadratic Functions (16:05)
 Define the following terms in your own words.
 Vertex
 Line of symmetry
 Parabola
 Minimum
 Maximum
 Without graphing, how can you tell if \begin{align*}y=ax^2+bx+c\end{align*} opens up or down?
Graph the following equations by making a table. Let \begin{align*}3 \le x \le 3\end{align*}. Determine the range of each equation.
 \begin{align*}y=2x^2\end{align*}
 \begin{align*}y=x^2\end{align*}
 \begin{align*}y=x^22x+3\end{align*}
 \begin{align*}y=2x^2+4x+1\end{align*}
 \begin{align*}y=x^2+3\end{align*}
 \begin{align*}y=x^28x+3\end{align*}
 \begin{align*}y=x^24\end{align*}
Does the graph of the parabola open up or down?
 \begin{align*}y=2x^22x3\end{align*}
 \begin{align*}y=3x^2\end{align*}
 \begin{align*}y=164x^2\end{align*}
Find the \begin{align*}x\end{align*}coordinate of the vertex of the following equations.
 \begin{align*}x^214x+45=0\end{align*}
 \begin{align*}8x^216x42=0\end{align*}
 \begin{align*}4x^2+16x+12=0\end{align*}
 \begin{align*}x^2+2x15=0\end{align*}
Graph the following functions by making a table of values. Use the vertex and \begin{align*}x\end{align*}intercepts to help you pick values for the table.
 \begin{align*}y=4x^24\end{align*}
 \begin{align*}y=x^2+x+12\end{align*}
 \begin{align*}y=2x^2+10x+8\end{align*}
 \begin{align*}y=\frac{1}{2} x^22x\end{align*}
 \begin{align*}y=x2x^2\end{align*}
 \begin{align*}y=4x^28x+4\end{align*}
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intercept of a parabola
As with linear equations, the intercepts of a parabola are where the graph intersects the axis. The value is zero at the intercepts.leading coefficient of a parabola
The variable in the equation is called the leading coefficient of the quadratic equation.minimums and maximums of a parabola
An equation of the form forms a parabola. If is positive, the parabola will open upward. The vertex will be a minimum. If is negative, the parabola will open downward. The vertex will be a maximum.symmetry of a parabola
A parabola can be divided in half by a vertical line. Because of this, parabolas have symmetry. The vertical line dividing the parabola into two equal portions is called the line of symmetry.vertex of a parabola
All parabolas have a vertex, the ordered pair that represents the bottom (or the top) of the curve. The line of symmetry always goes through the vertex. The vertex of a parabola is the ordered pair .Coefficient
A coefficient is the number in front of a variable.Dilation
To reduce or enlarge a figure according to a scale factor is a dilation.domain
The domain of a function is the set of values for which the function is defined.Horizontal shift
A horizontal shift is the result of adding a constant term to the function inside the parentheses. A positive term results in a shift to the left and a negative term in a shift to the right.Parabola
A parabola is the characteristic shape of a quadratic function graph, resembling a "U".quadratic function
A quadratic function is a function that can be written in the form , where , , and are real constants and .standard form
The standard form of a quadratic function is .Symmetry
A figure has symmetry if it can be transformed and still look the same.Vertex
The vertex of a parabola is the highest or lowest point on the graph of a parabola. The vertex is the maximum point of a parabola that opens downward and the minimum point of a parabola that opens upward.vertical axis
The vertical axis is also referred to as the axis of a coordinate graph. By convention, we graph the output variable on the axis.Image Attributions
Here you'll learn about the characteristics of a parabola, including its range and domain.