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Identifying Characteristics of Quadratic Functions

Like linear and exponential functions, quadratic functions are a unique type of functions that have specific qualities in common. Analyzing these functions in terms of their characteristics allows important information to be learned.
Concept

Quadratic Function

A quadratic function is a function of degree 2.2. That means that the highest exponent of the independent variable is 2.2. The simplest quadratic function is y=x2,y=x^2, and the graph of any quadratic function is a parabola.

Concept

Characteristics of Quadratic Functions

The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.

Concept

Direction

A parabola either opens upward or downward. This is called its direction.

Concept

Vertex

Because a parabola either opens upward or downward, there is always one point that is the absolute maximum or absolute minimum of the function. This point is called the vertex.

At the vertex, the function changes from increasing to decreasing, or vice versa.

Concept

Axis of Symmetry

All parabolas are symmetric, meaning there exists a line that divides the graph into two mirror images. For quadratic functions, that line is always parallel to the yy-axis, and is called the axis of symmetry.

The axis of symmetry always intersects the vertex of the parabola, and is written as a vertical line, where hh can be any real number.

x=hx=h


Concept

Zeros

Depending on its rule, a parabola can intersect the xx-axis at 0,0, 1,1, or 22 points. Since the function's value at an xx-intercept is always 0,0, these points are called zeros, or sometimes roots.

Concept

yy-intercept

Because all graphs of quadratic functions extend infinitely to the left and right, they each have a yy-intercept anywhere along the yy-axis.
fullscreen
Exercise

For the quadratic function y=x24x, y=x^2-4x, create a table of values to graph it. Then determine its direction, vertex, zeros, and axis of symmetry.

Show Solution
Solution
To begin, we'll use the function rule to create a table of values. Then, to graph the function, we'll plot the points from the table. Let's start with x=0.x=0.
y=x24xy=x^2-4x
y=024(0)y={\color{#0000FF}{0}}^2-4({\color{#0000FF}{0}})
y=0y=0
Thus, the point (0,0)(0,0) lies on the parabola. We can perform the same calculations for other xx-values around x=0.x=0.
xx x24xx^2-4x yy
-2{\color{#0000FF}{\text{-} 2}} (-2)24(-2)({\color{#0000FF}{\text{-} 2}})^2-4({\color{#0000FF}{\text{-} 2}}) 1212
-1{\color{#0000FF}{\text{-}1}} (-1)24(-1)({\color{#0000FF}{\text{-} 1}})^2-4({\color{#0000FF}{\text{-} 1}}) 55
0{\color{#0000FF}{0}} 024(0){\color{#0000FF}{0}}^2-4({\color{#0000FF}{0}}) 00
1{\color{#0000FF}{1}} 124(1){\color{#0000FF}{1}}^2-4( {\color{#0000FF}{1}}) -3\text{-} 3
2{\color{#0000FF}{2}} 224(2){\color{#0000FF}{2}}^2-4( {\color{#0000FF}{2}}) -4\text{-} 4

We'll plot these points on a coordinate plane.

We can start to see the left-hand side of the parabola. Let's add a few more xx-values to the table to determine a more complete shape.

xx x24xx^2-4x yy
3{\color{#0000FF}{3}} 324(3){\color{#0000FF}{3}}^2-4 ({\color{#0000FF}{3}}) -3\text{-} 3
4{\color{#0000FF}{4}} 424(4){\color{#0000FF}{4}}^2-4 ({\color{#0000FF}{4}}) 00
5{\color{#0000FF}{5}} 524(5){\color{#0000FF}{5}}^2-4 ({\color{#0000FF}{5}}) 55

We'll add these points to the coordinate system as well.

Looking at the points, we now see both sides of the parabola. We can connect the points with a smooth curve.

The graph can be used to describe the desired characteristics of the parabola.

direction:upwardaxis of symmetry:x=2vertex:(2,-4)zeros:(0,0) and (4,0)\begin{aligned} \textbf{direction} &: \text{upward} \\ \textbf{axis of symmetry} &: x=2\\ \textbf{vertex} &: (2,\text{-} 4)\\ \textbf{zeros} &: (0,0) \text{ and } (4,0) \end{aligned}

fullscreen
Exercise

Three quadratic functions are graphed in the coordinate plane.

For each graph, match it with the corresponding characteristics. direction:upward,downwardvertex:minimum,maximumvertex:(-2,2),(0,-6),(2,-4)axis of symmetry:x=-2,x=2,x=0y-intercept:y=-6,y=-2,   y=0,y=4zero:x=0, x=4,x=-6\begin{aligned} \textbf{direction:}& \quad \text{upward,} \quad \text{downward} \\ \textbf{vertex:}& \quad \text{minimum,} \quad \text{maximum} \\ \textbf{vertex:}& \quad (\text{-}2,2), \quad (0,\text{-}6), \quad (2,\text{-}4) \\ \textbf{axis of symmetry:}& \quad x=\text{-}2, \quad x=2, \quad x=0 \\ \textbf{$y$-intercept:}& \quad y=\text{-}6, \quad y=\text{-}2, \ \ \ y=0, \quad y=4\\ \textbf{zero:}& \quad x=0, \ \quad x=4, \quad x=\text{-}6 \end{aligned}

Show Solution
Solution

Instead of looking at each function separately, we'll look at the characteristics individually and summarize our findings in a table at the end.

Example

Direction and Min/Max

First, let's consider the direction of the parabolas. We can see that AA and CC open upward, and that BB opens downward. The direction of a parabola determines whether the vertex is a minimum or a maximum. Thus, the vertices of AA and CC are minimums while the vertex of BB is a maximum.

Example

Vertex

The vertex for each graph is found at the minimum or maximum of the function. For A,A, the vertex lies at (2,-4).(2,\text{-} 4). Similarly, BB's vertex lies at (-2,2)(\text{-} 2,2), and CC has its vertex at (0,-6).(0,\text{-}6).

Example

Axis of Symmetry

The axis of symmetry is the vertical line that intersects the vertex. Therefore, the axis of symmetry for graph AA is x=2,x=2, for BB it's x=-2x=\text{-}2 and for CC it is x=0.x=0.

Example

yy-intercept

The yy-intercepts are found where the parabolas intercept the yy-axis. AA has the yy-intercept y=0y=0, BB has y=-2y=\text{-} 2 and CC has y=-6y=\text{-} 6. One of the options, y=4,y=4, does not coincide with any graph.


Example

Zero

The zeros are found where the parabolas intercept the xx-axis. Then, function AA has zeros x=4x=4 and x=0x=0, which are two of the zeros given in the prompt. Neither function BB nor function CC has a zero at x=-6x=\text{-} 6.


Example

Summary

We summarize what we have learned about the characteristics of the three quadratic functions.

Function AA BB CC
direction upward downward upward
max/min minimum maximum minimum
vertex (2,-4)(2,\text{-} 4) (-2,2)(\text{-}2,2) (0,-6)(0,\text{-} 6)
axis of symmetry x=2x=2 x=-2x=\text{-}2 x=0x=0
yy-intercept y=0y=0 y=-2y=\text{-}2 y=-6y=\text{-}6
zeros x=0x=0 and x=4x=4 not applicable not applicable
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