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Quadratic Functions

Interpreting Quadratic Functions in Vertex Form

Quadratic function rules can be expressed in different ways. Each form is useful because it highlights different characteristics of the parabola. As the name implies, vertex form gives the vertex of the parabola.

Concept

Vertex Form

Vertex form is an algebraic format used to express quadratic function rules.

y=a(xh)2+k

In this form, a gives the direction of the parabola. When a>0, the parabola faces upward and when a<0, it faces downward. The vertex of the parabola lies at (h,k), and the axis of symmetry is x=h. Consider the graph of

From the graph, we can connect the following characterisitcs to the function rule.
Notice that although the factor in the function rule shows (x+4)2, h is actually equal to -4. This coincides with a horizontal translation of a quadratic function.

Method

Graphing a Quadratic Function in Vertex Form

When a quadratic function is written in vertex form, some characteristics of its graph can be identified.

y=a(xh)2+k
Direction Vertex Axis of Symmetry
(h,k) x=h
It is possible to graph a quadratic function using these characteristics. Consider the function y=(x2)24.

1

Identify and Plot the Vertex
To begin, identify the vertex (h,k) from the function rule.
Here, h=2 and Therefore, the vertex of the parabola is This point can be plotted on a coordinate plane.
The vertex (2,-4) on a coordinate plane

2

Draw the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is x=h, here it is x=2.

The line x=2 on a coordinate plane

3

Determine and Plot the y-intercept
Next, determine the y-intercept. Recall that for all y-intercepts, the x-coordinate is 0. Therefore, substitute x=0 into the function rule and solve for y.
y=(x2)24
y=(02)24
Evaluate right-hand side
y=(-2)24
y=44
y=0
The y-intercept of the parabola is (0,0). This point can be added to the graph.
y-intercept along with the vertex and axis of symmetry on a coordinate plane

4

Reflect the y-intercept Across the Axis of Symmetry
The axis of symmetry divides the parabola into two mirror images. Therefore, points on one side of the parabola can be reflected across the axis of symmetry.
y-intercept is reflected across the axis of symmetry

5

Draw the Parabola
Now, with three points plotted, the direction of the parabola can be seen. It appears that the parabola faces upward. Identify the value of a in the equation to see if this is correct.
Since a=1, which is greater than zero, it can be said that the parabola opens upward. To graph the quadratic function, connect the three points with a smooth curve.
The parabola is drawn through the three plotted points
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Exercise
The characteristics of two functions, f(x) and g(x), are described below. Determine which of the function rules could represent g and h.
Number Function rule
1 y=-(x1)2+3
2 y=(x1)23
3 y=-(x1)23
4 y=(x1)2+3
5 y=(x+1)2+3
6 y=(x+1)23
7 y=-(x+1)2+3
8 y=-(x+1)23
Show Solution
Solution
When a quadratic function is given in vertex form, y=a(xh)2+k, both the vertex and the axis of symmetry can be easily seen.
It is given that the axis of symmetry for g(x) is x=1 and for f(x) it is x=-1. Since the axis of symmetry is given as x=h, this gives h=1 for g and h=-1 for f. Thus, we can begin to write the function rules for g(x) and f(x) in vertex form as follows. Note that (x(-1)) becomes (x+1).
We can use the given vertices to determine the values of k. The vertex of g is (-1,-3). This gives k=-3. Similarly, f(x)'s vertex (1,3) gives k=3. Thus, we can write the rules as
Comparing these rules to the table above, it can be seen that rule number 2 and 3 can represent g(x) and 5 and 7 can represent f(x).
Number Function rule
2 y=(x1)23
3 y=-(x1)23
5 y=(x+1)2+3
7 y=-(x+1)2+3

There are two candidates remaining for each function. The rules with a=-1 correspond to a downward parabola, whereas the rules with a=1 correspond to an upward parabola. To determine if any option is completely correct, we'd need more information about the parabola, specifically its direction or if its vertex is a minimum or a maximum.

fullscreen
Exercise

Write the rule for the quadratic function that has the vertex (5,-1) and passes through the point (10,-11).

Show Solution
Solution
It's possible to write the rule of a quadratic function in vertex form (x)=a(xh)2+k using two points. Let's start with the vertex. In vertex form, (h,k) is the coordinate of the vertex. Knowing this, we can substitute h=5 and k=-1 into the rule.
f(x)=a(x5)21
Notice that the rule is incomplete. We can use the other point to determine the value of a. Since (10,-11) lies on the parabola, x=10 and y=-11 can be substituted into the rule. Then, we can solve for a.
y(x)=a(x5)21
-11=a(5)21
-11=25a1
-10=25a
-0.4=a
a=-0.4
Thus, we can write the complete function rule as
f(x)=-0.4(x5)21.
To verify that f passes through (5,1) and (10,-11), we can graph the parabola and mark the points.

Since the parabola passes through the given points, the created rule is correct.

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