{{ 'ml-label-loading-course' | message }}

{{ tocSubheader }}

{{ 'ml-toc-proceed-mlc' | message }}

{{ 'ml-toc-proceed-tbs' | message }}

An error ocurred, try again later!

Chapter {{ article.chapter.number }}

{{ article.number }}. # {{ article.displayTitle }}

{{ article.intro.summary }}

Show less Show more Lesson Settings & Tools

| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |

| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |

| {{ 'ml-lesson-time-estimation' | message }} |

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 $y-$intercept will then be understood. ### Catch-Up and Review

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

Explore

Consider the vertex form of a quadratic function $y=a(x−p)_{2}+q$ 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

Consider the intercept form of a quadratic function and its corresponding parabola. In the applet, adjust the parameters of each parabola's $x-$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

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.

$f(x)=ax_{2}+bx+c$

Here, $a,$ $b,$ and $c$ are real numbers with $a =0.$ 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 $y-$intercept, the axis of symmetry, and the vertex.

Direction of the Graph | Opens upward when $a>0$ |
---|---|

Opens downward when $a<0$ | |

$y-$intercept | $c$ |

Axis of Symmetry | $x=-2ab $ |

Vertex | $(-2ab ,f(-2ab ))$ |

Pop Quiz

Determine whether the given quadratic function is expressed in standard form.

Discussion

Given a quadratic function in standard form, some characteristics of its corresponding parabola can be determined. Consider an example quadratic function.
*expand_more*
*expand_more*
*expand_more*
*expand_more*
*expand_more*

$f(x)=x_{2}−4x+3 $

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

The axis of symmetry can be found by determining $a$ and $b$ in the form $ax_{2}+bx+c.$
The axis of symmetry of the function is the vertical line with equation $x=2.$

$f(x)=x_{2}−4x+3⇕f(x)=1x_{2}+(-4)x+3 $

In the given function, $a$ and $b$ are $1$ and $-4,$ respectively. Now, these values will be substituted into the formula for the equation of the axis of symmetry.
$x=-2ab $

SubstituteII

$a=1$, $b=-4$

$x=-2(1)-4 $

▼

Evaluate right-hand side

IdPropMult

Identity Property of Multiplication

$x=-2-4 $

RemoveNegFracAndNum

$-b-a =ba $

$x=24 $

CalcQuot

Calculate quotient

$x=2$

2

Determine and Plot the Vertex

The axis of symmetry intersects the parabola at its vertex. Therefore, $x=2$ is the $x-$coordinate of the vertex. To find the $y-$coordinate of the vertex, $x=2$ can be substituted into the function rule.
The vertex of the parabola is at $(2,-1).$

3

Determine and Plot the $y$-intercept

The $y-$intercept can be determined by using the constant term $c$ of the given function. In this case, $c$ is equal to $3.$ This means that the $y-$intercept occurs at $(0,3).$

4

Reflect the $y$-intercept Across the Axis of Symmetry

The axis of symmetry divides the graph into two *mirror images*. Therefore, the reflection of the $y-$intercept across the axis of symmetry is also on the parabola.

If the vertex of a quadratic function lies on the $y-$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 $y-$axis is the axis of symmetry.

5

Draw the Parabola

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 $a$ is $1,$ which is a positive number.

Pop Quiz

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 $2$ decimal places.

Example

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

$h(x)=-0.075x_{2}+1.35x $

Here, $x$ is the dog's horizontal distance from the jump spot and $h(x)$ 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 $7$ 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 $5-$foot-wide hole. How much further will the dog be able to jump as an adult?

a

b No

c $13$ feet

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.

a Consider the given quadratic function.

$h(x)=-0.075x_{2}+1.35x $

This function is written in standard form. To graph a quadratic function written in this form, there are five steps to follow. - Identify and graph the axis of symmetry
- Determine and plot the vertex
- Determine and plot the $y-$intercept
- Reflect the $y-$intercept in the axis of symmetry
- Draw the parabola

These steps will be done one at a time.

$h(x)=-0.075x_{2}+1.35x⇕h(x)=-0.075x_{2}+1.35x+0 $

Here, $a=-0.075,$ $b=1.35,$ and $c=0.$ The axis of symmetry is a vertical line with equation $x=2a-b .$
The axis of symmetry is the vertical line $x=9.$
$h(x)=-0.075x_{2}+1.35x$

Substitute

$x=9$

$h(9)=-0.075(9)_{2}+1.35(9)$

▼

Evaluate right-hand side

CalcPowProd

Calculate power and product

$h(9)=-0.075(81)+12.15$

MultNegPos

$(-a)b=-ab$

$h(9)=-6.075+12.15$

AddTerms

Add terms

$h(9)=6.075$

The $y-$intercept is given by the constant term $c$ of the function rule. In this case this term is equal to $0,$ which means that the $y-$intercept occurs at $(0,0)$ — the origin.

Now, another point that lies on the parabola can be found by reflecting the $y-$intercept in the axis of symmetry.

The third point that lies on the parabola is at $(18,0).$

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 $6.075$ feet and the height of the fence is $7$ 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 $5$ 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.

$18−5=13feet $

What a pup!
Discussion

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 | $y=a(x−h)_{2}+k$ | Expand $(x−h)_{2},$ distribute $a,$ and combine like terms. |

Intercept Form (also called Factored Form) |
$y=a(x−p)(x−q)$ | Multiply $a(x−p)(x−q)$ and combine like terms. |

Example

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 $Sk8$ sees this as an opportunity to make some big sales. SuTeKi $Sk8$ 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 | $y=-0.015(x−50)_{2}+37$ |

Sugoi Sketch | $y=-0.008(x−65)_{2}+28$ |

In these function rules, $x$ is the price in dollars and $y$ 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 $$80$ for each model. Then, compare the results with the exact values.

a **Standard Forms:**

Skateboard | Vertex Form | Standard Form |
---|---|---|

Sketchtastic | $y=-0.015(x−50)_{2}+37$ | $y=-0.015x_{2}+1.5x−0.5$ |

Sugoi Sketch | $y=-0.008(x−65)_{2}+28$ | $y=-0.008x_{2}+1.04x−5.8$ |

**Graphs:**

b See solution.

c **Estimation:**

Skateboard | Profit on Sales |
---|---|

Sketchtastic | $≈$23000$ |

Sugoi Sketch | $≈$26000$ |

**Comparison:** See solution.

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 $x=80.$ Then, substitute $x=80$ into the function rules to calculate the exact values.

a The quadratic functions that represent the data about the skateboard models are given in vertex form.

Vertex Form: $y=a(x−p)_{2}+q$ | |
---|---|

Sketchtastic | Sugoi Sketch |

$y=-0.015(x−50)_{2}+37$ | $y=-0.008(x−65)_{2}+28$ |

$y=-0.015(x−50)_{2}+37$

▼

Rewrite

ExpandNegPerfectSquare

$(a−b)_{2}=a_{2}−2ab+b_{2}$

$y=-0.015(x_{2}−2x(50)+50_{2})+37$

CalcPowProd

Calculate power and product

$y=-0.015(x_{2}−100x+2500)+37$

Distr

Distribute $-0.015$

$y=-0.015x_{2}+1.5x−37.5+37$

AddTerms

Add terms

$y=-0.015x_{2}+1.5x−0.5$

Sketchtastic | Sugoi Sketch | |
---|---|---|

Given Function | $y=-0.015(x−50)_{2}+37$ | $y=-0.008(x−65)_{2}+28$ |

Standard Form | $y=-0.015x_{2}+1.5x−0.5$ | $y=-0.008x_{2}+1.04x−5.8$ |

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 $y-$intercept.
- Reflect the $y-$intercept in the axis of symmetry.
- Draw the parabola.

The axis of symmetry is given by the equation $x=2a-b ,$ where $a$ is the coefficient of the $x_{2}-$term and $b$ is the linear coefficient. Use a calculator to evaluate the equations if necessary.

Axis of Symmetry: $x=2a-b $ | |
---|---|

$y=-0.015x_{2}+1.5x−0.5$ | $y=-0.008x_{2}+1.04x−5.8$ |

$x=2(-0.015)-1.5 $ | $x=2(-0.008)-1.04 $ |

$x=50$ | $x=65$ |

The axes of symmetry of the parabolas are the vertical lines with equations $x=50$ and $x=65.$

The vertices of the parabolas lie on the corresponding axes of symmetry. This means that $x=50$ and $x=65$ are the $x-$coordinates of the vertices. Now, these values will be substituted for $x$ into the function rules to find the $y-$coordinates.

$y=-0.015x_{2}+1.5x−0.5$ | $y=-0.008x_{2}+1.04x−5.8$ | |
---|---|---|

$x-$coordinate | $50$ | $65$ |

$y-$coordinate | $y=-0.015(50)_{2}+1.5(50)−0.5$ $⇕$ $y=37$ |
$y=-0.008(65)_{2}+1.04(65)−5.8$ $⇕$ $y=28$ |

Vertex | $(50,37)$ | $(65,28)$ |

The vertices can now be added to the graph.

The $y-$intercepts are given by the constant term of the function rules.

Function Rule | $y-$intercept |
---|---|

$y=-0.015x_{2}+1.5x−0.5$ | $-0.5$ |

$y=-0.008x_{2}+1.04x−5.8$ | $-5.8$ |

Next, another point that lies on each parabola can be found by reflecting the $y-$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, $x$ is the price in dollars and $y$ is the corresponding the profit on sales in thousands of dollars. Both functions reach their maximum values at their corresponding vertices.

The $x-$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 $$80,$ consider the graph of the functions once again. Since that price is given, a line corresponding to the price of $$80$ will be added to the graph.

From the graph, it can be seen that the profits of skateboards Sketchtastic and Sugoi Sketch are about $23$ and $26$ dollars in the thousands, respectively. To verify this, $x=80$ will be substituted in both function rules.

Sketchtastic | Sugoi Sketch | |
---|---|---|

$x$ | $80$ | $80$ |

Substitute | $-0.015(80)_{2}+1.5(80)−0.5$ | $-0.008(80)_{2}+1.04(80)−5.8$ |

Evaluate | $23.5$ | $26.2$ |

The estimated values are slightly less than the exact values.

Example

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 $80$ feet away. The ball follows a parabolic path and is caught at the height of $487 $ feet by a goalkeeper. The goalkeeper is standing $15$ feet in front of the goal.
### Answer

### Hint

### Solution

In the given scenario, the ball does not hit the goal's line. Instead, the goalkeeper, standing $15$ feet in front of the goal, catches the ball at a height of $487 $ feet.
Finally, the quadratic function that represents the ball's path can be written.

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 $50$ feet?

a $y=-0.005x(x−80)$

b **Standard Form:** $y=-0.005x_{2}+0.4x$

**Graph:**

c $8$ feet

d Yes, see solution.

a Illustrate the given situation on a coordinate plane. Mark the starting position of the ball and the goal's line on the $x-$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 $x$ is the function decreasing?

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 $80$ feet.
- If the goalkeeper does not catch the ball, it will hit the goal's line.
- The goalkeeper — standing $15$ feet in front of the goal — catches the ball at a height of $487 $ feet.

There are two $x-$intercepts. These are the starting point at $(0,0)$ and the goal's line at $(80,0).$ Therefore, to write the quadratic function representing the path of the ball, the intercept form can be used. The zeros of the parabola are $0$ and $80.$

$y=a(x−p)(x−q)$

SubstituteII

$p=0$, $q=80$

$y=a(x−0)(x−80)$

IdPropAdd

Identity Property of Addition

$y=ax(x−80)$

When the goalkeeper catches the ball, its horizontal distance from the starting point is $65$ feet. This means that the value of the function at $x=65$ is $y=487 .$ These values can be substituted into the intercept form to find the missing coefficient $a.$

$y=ax(x−80)$

SubstituteII

$x=65$, $y=487 $

$487 =a(65)(65−80)$

▼

Solve for $a$

SubTerms

Subtract terms

$487 =a(65)(-15)$

MultPosNeg

$a(-b)=-a⋅b$

$487 =a(-975)$

MixedToFrac

$acb =ca⋅c+b $

$839 =a(-975)$

DivEqn

$LHS/(-975)=RHS/(-975)$

$-97539/8 =a$

MoveNegDenomToFrac

Put minus sign in front of fraction

$-97539/8 =a$

DivFracD

$ba/c =b⋅ca $

$-780039 =a$

UseCalc

Use a calculator

$-0.005=a$

RearrangeEqn

Rearrange equation

$a=-0.005$

$y=ax(x−80)⇓y=-0.005x(x−80) $

b In Part A, a quadratic function written in intercept form has been obtained.

$y=-0.005x(x−80) $

To rewrite this function in standard form, the expression $-0.005x$ will be distributed.
$y=-0.005x(x−80)⇕y=-0.005x_{2}+0.4x $

The function 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 $y-$intercept
- Reflect the $y-$intercept in the axis of symmetry
- Draw the parabola