 Lesson

Previously, we looked at the features of quadratic functions and their graphs. We will now use this knowledge as we look at some physical situations that can be modeled by quadratic equations.

Perhaps the most common real-world occurrence of a quadratic equation is the motion of a falling object. When you throw a ball at an angle or hit it with a bat, the path it takes as it falls back to the ground is very close to a parabola (unless it is affected by a strong wind). A thrown ball follows the path of a parabola as it falls back to the ground.

Careful!

Quadratic functions of the form $y=ax^2+bx+c$y=ax2+bx+c that model objects falling under gravity may use $x$x to represent the horizontal position of the object. Other times, $x$x or $t$t may represent the time taken to reach a particular height $y$y. Make sure to pay attention to what the variables in the model represent!

The area formulas of various shapes can also be modeled by quadratic functions. For example, the area of a square of length $x$x is given by $A=x^2$A=x2, and the area of a circle of radius $r$r is given by $A=\pi r^2$A=πr2. There are many other situations that can be modeled by quadratic equations. The braking distance $d$d of a car is related to its speed $x$x by a quadratic equation of the form $d=ax^2+bx$d=ax2+bx. Braking distance is the distance a vehicle travels between the time the driver applies the brake and the time the vehicle stops.

Modeling physical phenomena

When using an equation to model a physical situation, the context is important when interpreting the results. For example, when modeling a physical situation with a parabola, the context can give meaning to the features of the parabola, such as the intercepts and the turning point.

If we think about the example of throwing a ball, we can model this using an equation of the form $y=ax^2+bx+c$y=ax2+bx+c, where $y$y is the height of the ball at time $x$x. In this model:

• the $y$y-intercept of the parabola represents the height that the ball was thrown from,
• the turning point represents the maximum height that the ball reaches, and
• the $x$x-intercept represents how long it took to reach the ground.

When modeling the path of a ball using a parabola, we can highlight the important features, as shown below. We also restrict the curve to positive times and heights, because only these values exist within this context. The parts of the parabola drawn with dashes do not model the physical situation of throwing a ball.

When we use mathematics to model real contexts, we should always consider any practical limitations. For example, the area of a square only makes sense when the side length is positive (that is, for $x>0$x>0), even though the parabola $A=x^2$A=x2 that models it can take negative values.

Practice questions

Question 1

The formula for the surface area of a sphere is $S=4\pi r^2$S=4πr2, where $r$r is the radius in centimeters.

1. Fill in the following table of values for $S=4\pi r^2$S=4πr2, giving your answers correct to two decimal places.

 $r$r $S$S $1$1 $2$2 $3$3 $4$4 $5$5 $6$6 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Choose the graph that correctly shows $S=4\pi r^2$S=4πr2.

A

B

C

D

A

B

C

D
3. Use the graph from part (b) to approximate the surface area of a sphere of radius $5.5$5.5 cm.

$S=350$S=350 cm2

A

$S=410$S=410 cm2

B

$S=380$S=380 cm2

C

$S=360$S=360 cm2

D

$S=350$S=350 cm2

A

$S=410$S=410 cm2

B

$S=380$S=380 cm2

C

$S=360$S=360 cm2

D
4. Now use the graph from part (b) to approximate the radius of a sphere that has a surface area of $804$804 cm2.

$r=10$r=10 cm

A

$r=8$r=8 cm

B

$r=6$r=6 cm

C

$r=9$r=9 cm

D

$r=10$r=10 cm

A

$r=8$r=8 cm

B

$r=6$r=6 cm

C

$r=9$r=9 cm

D

Question 2

A ball is thrown into the air at an angle. The height $y$y (in meters) of the ball at time $x$x (in seconds) is modeled by the equation $y=20x-5x^2$y=20x5x2.

The graph of this relationship is shown below.

1. How long does it take for the ball to return to the ground?

2. What does the $y$y-value of the turning point represent?

The time it takes for the ball to reach its maximum height.

A

The maximum height reached by the ball.

B

The time it takes for the ball to reach the ground.

C

The starting height of the ball.

D

The time it takes for the ball to reach its maximum height.

A

The maximum height reached by the ball.

B

The time it takes for the ball to reach the ground.

C

The starting height of the ball.

D
3. For what $x$x-values does this model make sense?

From $x=0$x=0 to $x=4$x=4.

A

From $x=0$x=0 to $x=20$x=20.

B

All $x$x values greater than $0$0.

C

From $x=0$x=0 to $x=4$x=4.

A

From $x=0$x=0 to $x=20$x=20.

B

All $x$x values greater than $0$0.

C