Introduction to Parabolas

1. Which parabola has exactly one $x$-intercept?

Choose 1 answer:

A.

B.

C.

2. Graph a parabola whose vertex is at $(3, 5)$ with $y$-intercept at $y=1$.

3. Mark the vertex of the parabola.

4. Graph a parabola whose $x$-intercepts are at $x = -3$ and $x = 5$ and whose minimum value is $y=-4$.

1. C.

.Let's review each option.

This parabola intersects the $x$-axis at two points.

This parabola doesn't intersect the $x$-axis at all! (Yes, this can happen with parabolas.

This parabola touches the $x$-axis with its vertex. Therefore, it has exactly one $x$-intercept.

Note that whenever a parabola has exactly one $x$-intercept, that intercept will be the vertex, and the parabola will be touching the $x$-axis instead of crossing it.

2. We start with a vertex. This vertex is either the minimum or the maximum point of the parabola.

Since the $y$-intercept is below the vertex, we know the parabola opens down.

This is the parabola.

3. The vertex of a parabola is the lowest or highest point of the parabola (depending on whether the parabola opens up or down).

The vertex is also the parabola's intersection with its axis of symmetry.

The vertex of our parabola is at $(2, -4)$.

4. The first thing we have to do in order to graph the parabola is find its vertex.

We are given that the parabola's minimum value is $y = -4$, so we know the vertex's $y$-coordinate is $-4$.

We can find the vertex's $x$-coordinate by taking the average of the two $x$-intercepts:

$\frac{(-3)+(5)}{2}=1$

So the vertex is at $(1,-4)$.

In order to graph, we need one more point on the parabola. We can use one of the given $x$-intercepts, like $(-3,0)$.

1. Biologists introduced a population of bacteria to a test environment. The function $f$ models the population size (in thousands) as a function of time (in days) after introduction.

Plot the point on the graph of $f$ that corresponds to when there are no more bacteria in the population.

2. A baby bird jumps from a tree branch and flutters to the ground. The function $f$ models the bird's height (in meters) above the ground as a function of time (in seconds) after jumping.

Plot the point on the graph of $f$ that corresponds to the bird's height above the ground when it jumped.

3.Cassie hit a golf ball. The function $f$ models the height of the ball above the ground (in meters) as a function of time (in seconds) after Cassie hit it.

Plot the point on the graph of $f$ that corresponds to the ball's maximum height.

4. Mia kicked a football. The function $f$ models the height of the ball above the ground (in meters) as a function of time (in seconds) after Mia kicked it.

Plot the point on the graph of $f$ that corresponds to when the ball hit the ground.

1.

Which feature?

Let's think about what each feature of a parabola shows about the function it represents:

Which feature corresponds to when there are no more bacteria in the population?

The $x$-intercept $(4,0)$ shows that at $4$ days, the population size is $0$.

Answer

The $x$-intercept $(4,0)$ corresponds to when there are no more bacteria in the population.

2.

Which feature?

Let's think about what each feature of a parabola shows about the function it represents:

Which feature corresponds to the bird's height above the ground when it jumped?

The $y$-intercept shows that at the moment the bird jumped, it was $12$ meters above the ground.

Answer

The $y$-intercept $(0,12)$ corresponds to the bird's height above the ground when it jumped.

3.

Which feature?

Let's think about what each feature of a parabola shows about the function it represents:

Which feature corresponds to the ball's maximum height?

The vertex shows that $4$ seconds after Cassie hit it, the ball reached a maximum height of $80$ meters above the ground.

Answer

The vertex $(4,80)$ corresponds to the ball's maximum height.

4.

Which feature?

Let's think about what each feature of a parabola shows about the function it represents:

Which feature corresponds to when the ball hit the ground?

The $x$-intercept $(4,0)$ shows that $4$ seconds after Mia kicked it, the ball is 000 meters above the ground.

Answer

The $x$-intercept $(4,0)$ corresponds to when the ball hit the ground.

1. The function $g$ models the number of mosquitoes (in millions of mosquitoes) in a certain area as a function of rainfall (in centimeters) in that area.

Which of these statements are true?

Choose all answers that apply:

A. There's the largest possible number of mosquitoes when rainfall is about $2 \text { cm }$.

B. There's the largest possible number of mosquitoes when rainfall is about $4 \text{ cm }$.

C. There are no mosquitoes when there is no rainfall.

D. There are no mosquitoes when rainfall is about $4 \text{ cm}$.

2. Simon has a certain length of fencing to enclose a rectangular area. The function $A$ models the rectangle's area (in square meters) as a function of its width (in meters).

Which of these statements are true?

Choose all answers that apply:

A. Greater width relates to greater area as long as the width is less than $10\text{ m}$.

B. Greater width relates to greater area as long as the width is more than $10\text{ m}$.

C. To get the greatest area, the width should be $10\text{ m}$.

D. To get the greatest area, the width should be $20\text{ m}$.

3. A certain company's main source of income is a mobile app. The function $h$ models the company's annual profit (in millions of dollars) as a function of the price they charge for the app (in dollars).

Which of these statements are true?

Choose all answers that apply:

A. If the company doesn't charge for the app, they make $0$ profit.

B. If the company doesn't charge for the app, they lose about $10$ million dollars.

C. To make the greatest profit, the company should charge $7$ dollars for the app.

D. To make the greatest profit, the company should charge $40$ dollars for the app.

4. Sarah kicked a ball in the air. The function $f$ models the height of the ball (in meters) as a function of time (in seconds) after Sarah kicked it.

Which of these statements are true?

Choose all answers that apply:

A. The ball moved upwards for about $3.5\text{ s}$.

B. The ball started moving upwards after about $1.75\text{ s}$.

C. The ball hit the ground after about $3.5\text{ s}$.

D. The ball hit the ground after about $1.75\text{ s}$.

1. A, C and D

There are two pairs of statements. Each pair is about a different property:

1. The first pair is about the amount of rainfall when there's the largest possible number of mosquitoes. The $x$-coordinate of the vertex represents this.
2. The second pair is about the amount of rainfall when there are no mosquitoes. The parabola's $x$-intercepts represent this.

The amount of rainfall when there's the largest possible number of mosquitoes

Let's look at the $x$-coordinate of the vertex.

We can see that the $x$-coordinate at the vertex is about $2$.

The conditions for no mosquitoes

Let's look at the $x$-intercepts.

We can see that the parabola intersects the $x$-axis both when rainfall is $0$ and when it's $4 \text{ cm}$.

Answers

There are three correct statements:

• There's the largest possible number of mosquitoes when rainfall is about $2 \text{ cm}$.
• There are no mosquitoes when there is no rainfall.
• There are no mosquitoes when rainfall is about $4 \text{ cm}$.

2. A and C

There are two pairs of statements. Each pair is about a different property:

1. The first pair is about the conditions for this behavior: "Greater width relates to greater area." This is a description of an increasing function.
2. The second pair is about the width when the area is the greatest possible. The \x\)-coordinate of the vertex represents this.

The conditions for "Greater width relates to greater area"

Let's see where the function increases.

We can see that the function increases when the width is less than $10\text{ m}10$.

The width when the area is the greatest possible

Let's look at the $x$-coordinate of the vertex.

We can see that the width at the vertex is about $10\text{ m}$.

Answers

There are two correct statements:

• Greater width relates to greater area as long as the width is less than $10\text{ m}$.
• To get the greatest area, the width should be $10\text{ m}$.

3. B and C

There are two pairs of statements. Each pair is about a different property:

1. The first pair is about the company's profit if they don't charge for their app. The $/y$-intercept represents this.
2. The second pair is about the app price that will bring the company the greatest profit. The $x$-coordinate of the vertex represents this.

The company's profit if they don't charge for their app

Let's look at the $y$-intercept.

We can see that the profit at the $y$-intercept is about $-10$ million dollars.

The app price that will bring the company the greatest profit

Let's look at the $x$-coordinate of the vertex.

We can see that the price at the vertex is about $7$ dollars.

Answers

There are two correct statements:

• If the company doesn't charge for the app, they lose about $10$ million dollars.
• To make the greatest profit, the company should charge $7$ dollars for the app.

4. C

There are two pairs of statements. Each pair is about a different property:

1. The first pair is about the time period when the ball moved upwards. This is a description of an increasing function.
2. The second pair is about the time it took the ball to hit the ground. The second $x$-intercept represents this.

The time period when the ball moved upwards

Let's see where the function increases.

We can see that the function increases until about $1.75\text{ s}$, not after it, and not for $3.5\text{ s}$.

The time it took the ball to hit the ground

Let's look at the $x$-intercept.

We can see that time at the $x$-intercept is about $3.5\text{ s}$.

Answers

There is only one correct statement:

• The ball hit the ground after about $3.5\text{ s}$.