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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.
Explore
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.
Explore
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.
Discussion
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. The standard form of the function 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 | |
| intercept | |
| Axis of Symmetry | |
| Vertex |
Pop Quiz
Is the Quadratic Function Written in Standard Form?
Determine whether the given quadratic function is expressed in standard form.
Discussion
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
expand_more 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.
The axis of symmetry of the function is the vertical line with equation 
▼
Evaluate right-hand side
2
Determine and Plot the Vertex
expand_more 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.
The vertex of the parabola is at 
▼
Evaluate right-hand side
3
Determine and Plot the -intercept
expand_more 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
expand_more 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
expand_more 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.
Pop Quiz
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.
Example
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!
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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. 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?
Answer
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.
- Identify and graph the axis of symmetry
- Determine and plot the vertex
- Determine and plot the intercept
- Reflect the intercept in the axis of symmetry
- 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.
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 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.
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.
Discussion
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 intercept forms of a quadratic function can always be rewritten in standard form.
| Form | Equation | How to Rewrite? |
|---|---|---|
| Vertex Form | Expand distribute and combine like terms. | |
| Intercept Form (also called Factored Form) |
Multiply and combine like terms. |
Example
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.
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.
Answer
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 |
▼
Rewrite
| 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.
- Identify and graph the axis of symmetry.
- Determine and plot the vertex.
- Determine and plot the intercept.
- Reflect the intercept in the axis of symmetry.
- 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.
Example
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.
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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?
Answer
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.
▼
Solve for
SubTerms
Subtract terms
MoveNegDenomToFrac
Put minus sign in front of fraction
UseCalc
Use a calculator
RearrangeEqn
Rearrange equation
b In Part A, a quadratic function written in intercept form has been obtained.
- Identify and graph the axis of symmetry
- Determine and plot the vertex
- Determine and plot the intercept
- Reflect the intercept in the axis of symmetry
- Draw the parabola
The intercept is given by the constant term of the function rule, which in this case is Therefore, the intercept is at the origin
Now, another point that lies on the parabola can be found by reflecting the intercept in the axis of symmetry.
The third point lies at Finally, the three points will be connected with a smooth curve to graph the parabolic shape. Since the function represents height of the ball, negative values of the function will not be considered.
c To determine the maximum height of the ball, the maximum of the given function will be considered.
A parabola that opens downward reaches its maximum at the vertex. In Part B, it was obtained that the vertex of the given function is at Therefore, the maximum height of the ball is feet.
d Consider the graph of the given function once again. The horizontal distance from the kicker — who is placed at the origin — will be marked in the graph.
Recall that the coordinate of the vertex is Since the given distance is greater than it can be said that the ball is already falling at
If the goalie understands this math concept, and can apply it in the heat of a soccer match, they are surely going to be an amazing player and maybe some day a sports scientist. What exciting times in what Zain thought was a sleepy town!
Pop Quiz
Rewriting a Quadratic Function in Standard Form
The quadratic functions are given in either intercept or vertex form. State the coefficients and of the standard form
Closure
Graphing Quadratic Functions in Different Forms
The three most common forms of a quadratic function are standard form, intercept form, and vertex form.
| Form | Formula |
|---|---|
| Standard | |
| Intercept | |
| Vertex |
In these forms, is not equal to Sometimes, the function rule of a quadratic function is not known. Instead of the function rule, the intercept and at least two more points that lie on the parabola may be given. In this case, the function can also be graphed using its standard form. The coefficients of the standard form can be found by following four steps.
- Use the intercept to find the constant term
- Substitute the coordinates of the points into the general standard form for and
- Solve the obtained system of equations for and
- Write the standard form of the function
An example will be considered.
Example
Consider the intercept of a parabola and two of its points. Now, the standard form of the function will be found by following the mentioned steps.1
Use the intercept to Find
expand_more The intercept occurs at This means that is equal to
2
Substitute the Coordinates of the Points
expand_more Next, the coordinates of the given points will be substituted for and into the obtained partial equation. Each point will be considered one at a time starting with
Similarly, the coordinates of the second point will be substituted.
The obtained equations are and
▼
Simplify
▼
Simplify
CalcPow
Calculate power
CommutativePropMult
Commutative Property of Multiplication
RearrangeEqn
Rearrange equation
3
Solve the System of Equations
expand_more Now, consider the system of equations formed by the obtained equations.
The system can be solved by using the Substitution Method. For simplicity, in this case variable will be isolated in Equation (I).
Now, Equation (II) will be solved for
▼
Solve for
AddTerms
Add terms
MoveNegDenomToFrac
Put minus sign in front of fraction
CalcQuot
Calculate quotient
4
Write the Standard Form
expand_more Finally, the standard form can be written.
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