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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.


Vertex Form

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

In this form, gives the direction of the parabola. When the parabola faces upward and when it faces downward. The vertex of the parabola lies at and the axis of symmetry is 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 is actually equal to This coincides with a horizontal translation of a quadratic function.


Graphing a Quadratic Function in Vertex Form

Writing quadratic functions in vertex form, is advantageous because it clearly presents three characteristics of the function. It is possible to graph a quadratic function using its characteristics. Consider the function


Identify and plot the vertex

To begin, identify the vertex, from the function rule. Since the rule is Thus, the vertex is Next, plot the vertex on a coordinate plane.


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 here it is


Determine and plot the -intercept
Next, determine the -intercept. Recall that for all -intercepts, the -coordinate is Thus, substitute into the function rule and solve for
Substitute for and evaluate
Thus, the -intercept of the parabola is This point can be added to the graph.


Reflect the -intercept across the axis of symmetry

The axis of symmetry divides the parabola into two mirror images. Thus, points on one side of the parabola can be reflected across the axis of symmetry. Notice that the -intercept is units away from the axis of symmetry.

There exists another point directly across from the -intercept that is also units from the axis of symmetry.


Draw the parabola

Now, with three points plotted, the general shape of the parabola can be seen. It appears that the parabola faces upward. Since in the given function rule, this should be expected. To draw the parabola, connect the points with a smooth curve.


The characteristics of two functions, and are described below. Determine which of the function rules could represent and

Number Function rule
Show Solution

When a quadratic function is given in vertex form, , both the vertex and the axis of symmetry can be easily seen. It is given that the axis of symmetry for is and for it is Since the axis of symmetry is given as this gives for and for Thus, we can begin to write the function rules for and in vertex form as follows. Note that becomes We can use the given vertices to determine the values of The vertex of is This gives Similarly, 's vertex gives Thus, we can write the rules as Comparing these rules to the table above, it can be seen that rule number and can represent and and can represent

Number Function rule

There are two candidates remaining for each function. The rules with correspond to a downward parabola, whereas the rules with 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.


Write the rule for the quadratic function that has the vertex and passes through the point .

Show Solution
It's possible to write the rule of a quadratic function in vertex form using two points. Let's start with the vertex. In vertex form, is the coordinate of the vertex. Knowing this, we can substitute and into the rule. Notice that the rule is incomplete. We can use the other point to determine the value of Since lies on the parabola, and can be substituted into the rule. Then, we can solve for
Thus, we can write the complete function rule as To verify that passes through and 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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