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This lesson explains what the standard form of a quadratic function is and how to draw its graph. How to determine various characteristics of a quadratic function such as the axis of symmetry, the vertex, and the intercept will then be understood.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Expanding the Vertex Form of a Quadratic Function

Consider the vertex form of a quadratic function and its corresponding parabola. In the applet, adjust the parameters of each parabola's vertex. Investigate how the expanded form of the equation is rewritten according to the new parameters. Since the obtained function is equivalent to the given function in vertex form, the parabola also corresponds to the obtained function.

## Expanding the Intercept Form of a Quadratic Function

Consider the intercept form of a quadratic function and its corresponding parabola. In the applet, adjust the parameters of each parabola's intercepts. Investigate how the expanded form of the equation is rewritten according to the new parameters. Since the obtained function is equivalent to the given function in intercept form, the parabola also corresponds to the obtained function.

## Standard Form of a Quadratic Function

Besides the intercept and the vertex forms, another essential and common form of a quadratic function is its standard form.

The standard form of a quadratic function is a quadratic function written in a specific format.

Here, and are real numbers with The term with the highest degree — the quadratic term — is written first. Then, the linear term followed by the constant term are written. The standard form can be used to determine the direction of the parabola, the intercept, the axis of symmetry, and the vertex.

 Direction of the Graph Opens upward when Opens downward when

## Is the Quadratic Function Written in Standard Form?

Determine whether the given quadratic function is expressed in standard form. ## Graphing a Quadratic Function in Standard Form

Given a quadratic function in standard form, some characteristics of its corresponding parabola can be determined. Consider an example quadratic function.
To draw the graph of the function written in standard form, there are five steps to follow.
1
Identify and Graph the Axis of Symmetry
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The axis of symmetry can be found by determining and in the form
In the given function, and are and respectively. Now, these values will be substituted into the formula for the equation of the axis of symmetry.
Evaluate right-hand side
The axis of symmetry of the function is the vertical line with equation 2
Determine and Plot the Vertex
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The axis of symmetry intersects the parabola at its vertex. Therefore, is the coordinate of the vertex. To find the coordinate of the vertex, can be substituted into the function rule.
Evaluate right-hand side
The vertex of the parabola is at 3
Determine and Plot the -intercept
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The intercept can be determined by using the constant term of the given function. In this case, is equal to This means that the intercept occurs at 4
Reflect the -intercept Across the Axis of Symmetry
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The axis of symmetry divides the graph into two mirror images. Therefore, the reflection of the intercept across the axis of symmetry is also on the parabola. If the vertex of a quadratic function lies on the axis, then any point that lies on the graph other than the vertex should be found and reflected across the axis of symmetry. In this case, the axis is the axis of symmetry.

5
Draw the Parabola
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Finally, connect the points with a smooth curve to graph the parabola. The graph of the function opens upward. This is expected, since the value of is which is a positive number.

## Determining the Vertex of a Quadratic Function Written in Standard Form

Find the coordinates of the vertex of the parabola that corresponds to the quadratic function written in standard form. If necessary, round the answer to decimal places. ## Will a German Shepherd Jump Over the Fence?

Zain lives in a quiet rural town where it seems not a lot happens. He decides to adopt a German shepherd puppy. After a few short weeks the puppy is growing so fast, they are afraid that soon the pup will be able to jump over the fence around their garden! External credits: @brgfx
Zain surfs the net and finds that a researcher has actually modeled the path of an adult German shepherd's jump! The path Zain found is a parabolic path modeled by a quadratic function.
Here, is the dog's horizontal distance from the jump spot and is the height of that jump. Both values are given in feet.
a Draw the function's graph.
b If the height of the fence is feet, will their dog, when it is an adult, manage to jump over the fence?
c Zosia and Zain made one more observation about their puppy's mad hops. The puppy can already jump, horizontally, over a foot-wide hole. How much further will the dog be able to jump as an adult?

a b No
c feet

### Hint

a Start by determining the axis of symmetry and the vertex of the parabola. Then, find two more points that lie on the curve.
b Consider the coordinates of the parabola's vertex.
c Use the graph from Part A.

### Solution

a Consider the given quadratic function.
This function is written in standard form. To graph a quadratic function written in this form, there are five steps to follow.
1. Identify and graph the axis of symmetry
2. Determine and plot the vertex
3. Determine and plot the intercept
4. Reflect the intercept in the axis of symmetry
5. Draw the parabola

These steps will be done one at a time.

### The Axis of Symmetry

To find the equation of the axis of symmetry, the coefficients and should first be found. Consider the given function.
Here, and The axis of symmetry is a vertical line with equation
Evaluate right-hand side
The axis of symmetry is the vertical line ### The Vertex

The vertex of the function lies on the axis of symmetry. This means that the vertex's coordinate is Now, to find the coordinate, will be substituted into the function rule.
Evaluate right-hand side
The vertex of the function is at ### The intercept

The intercept is given by the constant term of the function rule. In this case this term is equal to which means that the intercept occurs at — the origin. ### Reflecting the intercept

Now, another point that lies on the parabola can be found by reflecting the intercept in the axis of symmetry. The third point that lies on the parabola is at

### Drawing the Parabola

Finally, the points will be connected with a smooth curve to graph the parabolic shape. Since the function represents a dog's jump, negative values of the function will not be included. b To determine whether the adult dog will be able to jump over the fence, the fence will be drawn in the same coordinate plane. Assume that the fence is placed at the axis of symmetry, where the height of the jump would the greatest. Since the greatest height that the dog will reach is feet and the height of the fence is feet, the dog will not be able to jump over the fence.
c It is known that the dog can jump over a hole whose width is feet. Consider the graph of the function once again and note the horizontal distance that the dog will be able to jump as an adult. The maximum horizontal distance that the dog will be able to jump corresponds to the distance between the intercepts. This distance is feet. To find how much further the dog will be able to jump, the difference between and must be calculated.
What a pup!

## Rewriting a Quadratic Function in Standard Form

Depending on the conditions, it is convenient to rewrite a quadratic function given in intercept or vertex form in its standard form.

Both the vertex and the intercept form of a quadratic function can always be written in standard form.

Form Equation How to Rewrite?
Vertex Form Expand distribute and combine like terms.
Intercept Form Multiply and combine like terms.

## Graphing Quadratic Functions to Represent Profit

The mayor of Zain's sleepy town wants to do something exciting for the community. Lucky her, she noticed an increasing interest in people taking up skateboarding, so she knew that building a new skate ramp would be a huge hit. She was right! Now a local skateboard company named SuTeKi sees this as an opportunity to make some big sales. SuTeKi investigates the profit of its two coolest models, Sketchtastic and Sugoi Sketch. They collect data and find a relationship between the price and the profit on sales of both skateboards. The following quadratic functions represent their results.

Skateboard Relationship
Sketchtastic
Sugoi Sketch

In these function rules, is the price in dollars and is the corresponding profit in thousands of dollars.

a Rewrite the quadratic functions in standard form and graph them in the same coordinate plane.
b What does the vertex of each parabola represent in this context?
c Using the graph, estimate the profit on sales if they set the price at for each model. Then, compare the results with the exact values.

a Standard Forms:
Skateboard Vertex Form Standard Form
Sketchtastic
Sugoi Sketch

Graphs: b See solution.
c Estimation:
Skateboard Profit on Sales
Sketchtastic
Sugoi Sketch

Comparison: See solution.

### Hint

a To rewrite the quadratic functions in standard form, start by expanding the squared expression. To draw a parabola, determine the axis of symmetry and the vertex. Then, find two more points on the curve.
b The vertex of a parabola represents the maximum or minimum value of the function.
c To estimate the profit, look at the values of the function at Then, substitute into the function rules to calculate the exact values.

### Solution

a The quadratic functions that represent the data about the skateboard models are given in vertex form.
Vertex Form:
Sketchtastic Sugoi Sketch
The given functions can be rewritten in standard form by expanding the squared expression. First, the function corresponding to the Sketchtastic model will be rewritten.
Rewrite
By following the same procedure, the second function that represents Sugoi Sketch can also be rewritten in standard form.
Sketchtastic Sugoi Sketch
Given Function
Standard Form

The functions can now be graphed using the standard form. To do so, there are five steps to follow.

1. Identify and graph the axis of symmetry.
2. Determine and plot the vertex.
3. Determine and plot the intercept.
4. Reflect the intercept in the axis of symmetry.
5. Draw the parabola.

The axis of symmetry is given by the equation where is the coefficient of the term and is the linear coefficient. Use a calculator to evaluate the equations if necessary.

Axis of Symmetry:

The axes of symmetry of the parabolas are the vertical lines with equations and The vertices of the parabolas lie on the corresponding axes of symmetry. This means that and are the coordinates of the vertices. Now, these values will be substituted for into the function rules to find the coordinates.

coordinate
coordinate

Vertex

The vertices can now be added to the graph. The intercepts are given by the constant term of the function rules.

Function Rule intercept

Next, another point that lies on each parabola can be found by reflecting the intercept in the axis of symmetry. Finally, the corresponding points will be connected with a smooth curve to graph the parabolic shapes. b In the given functions, is the price in dollars and is the corresponding the profit on sales in thousands of dollars. Both functions reach their maximum values at their corresponding vertices. The coordinate of each vertex is the price of the corresponding skateboard at which the profit of the company is maximized. This means that the vertex represents the optimal price and its corresponding profit. The company should use these values to make the highest return on sales.

c To estimate the profit of the skateboards if the price is set at consider the graph of the functions once again. Since that price is given, a line corresponding to the price of will be added to the graph. From the graph, it can be seen that the profits of skateboards Sketchtastic and Sugoi Sketch are about and dollars in the thousands, respectively. To verify this, will be substituted in both function rules.

Sketchtastic Sugoi Sketch
Substitute
Evaluate

The estimated values are slightly less than the exact values.

## Using a Quadratic Function to Model the Trajectory of a Kicked Ball

Zain's sleepy town is becoming more exciting these days. Two rival soccer teams, feeling the town's energy, meet at the park. The two captains begin a duel of sorts, and a soccer ball is kicked from the ground level towards a goal feet away. The ball follows a parabolic path and is caught at the height of feet by a goalkeeper. The goalkeeper is standing feet in front of the goal. External credits: Jeffrey F Lin
a In its intercept form, write the equation of the quadratic function that represents the path of the ball, assuming that it would hit the goal's line.
b Rewrite the obtained function in standard form. Then, graph the function.
c What is the maximum height of the ball?
d Is the ball already falling when its horizontal distance from the kicker is feet?

a
b Standard Form:

Graph: c feet
d Yes, see solution.

### Hint

a Illustrate the given situation on a coordinate plane. Mark the starting position of the ball and the goal's line on the axis. Then, use intercept form of a quadratic function to find the function rule.
b Rewrite the function obtained in Part A by expanding the function rule. Next, determine the axis of symmetry and the vertex of the given function. Then, find two more points that lie on the parabola.
c At what point does a parabola reach its maximum or minimum value?
d Consider the graph of the function. For what values of is the function decreasing?

### Solution

a To write the quadratic function that represents the path of the ball, consider all the information given.
• A player kicks the ball from a distance of feet.
• If the goalkeeper does not catch the ball, it will hit the goal's line.
• The goalkeeper — standing feet in front of the goal — catches the ball at a height of feet.
The given situation can be illustrated on a coordinate plane. Assume that the starting position of the ball is at the origin. The axis represents the horizontal distance from the starting point and the axis represents the height of the ball. There are two intercepts. These are the starting point at and the goal's line at Therefore, to write the quadratic function representing the path of the ball, the intercept form can be used. The zeros of the parabola are and
In the given scenario, the ball does not hit the goal's line. Instead, the goalkeeper, standing feet in front of the goal, catches the ball at a height of feet. When the goalkeeper catches the ball, its horizontal distance from the starting point is feet. This means that the value of the function at is These values can be substituted into the intercept form to find the missing coefficient
Solve for
Finally, the quadratic function that represents the ball's path can be written.
b In Part A, a quadratic function written in intercept form has been obtained.
To rewrite this function in standard form, the expression will be distributed.
The function can now be graphed using the standard form. To do so, there are five steps to follow.
1. Identify and graph the axis of symmetry
2. Determine and plot the vertex
3. Determine and plot the intercept
4. Reflect the intercept in the axis of symmetry
5. Draw the parabola
The axis of symmetry is the vertical line with equation where is the coefficient of the term and is the linear coefficient.
Evaluate right-hand side