11. Modeling with Functions

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).

Careful!

Quadratic functions of the form $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`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`=`x`2, and the area of a circle of radius $r$`r` is given by $A=\pi r^2$`A`=π`r`2.

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`=`a``x`2+`b``x`.

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`=`a``x`2+`b``x`+`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.

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`=`x`2 that models it can take negative values.

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

Fill in the following table of values for $S=4\pi r^2$

`S`=4π`r`2, giving your answers correct to two decimal places.$r$ `r`$1$1 $2$2 $3$3 $4$4 $5$5 $6$6 $S$ `S`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Choose the graph that correctly shows $S=4\pi r^2$

`S`=4π`r`2.Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DUse the graph from part (b) to approximate the surface area of a sphere of radius $5.5$5.5 cm.

$S=350$

`S`=350 cm^{2}A$S=410$

`S`=410 cm^{2}B$S=380$

`S`=380 cm^{2}C$S=360$

`S`=360 cm^{2}D$S=350$

`S`=350 cm^{2}A$S=410$

`S`=410 cm^{2}B$S=380$

`S`=380 cm^{2}C$S=360$

`S`=360 cm^{2}DNow use the graph from part (b) to approximate the radius of a sphere that has a surface area of $804$804 cm

^{2}.$r=10$

`r`=10 cmA$r=8$

`r`=8 cmB$r=6$

`r`=6 cmC$r=9$

`r`=9 cmD$r=10$

`r`=10 cmA$r=8$

`r`=8 cmB$r=6$

`r`=6 cmC$r=9$

`r`=9 cmD

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`=20`x`−5`x`2.

The graph of this relationship is shown below.

Loading Graph...

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

What does the $y$

`y`-value of the turning point represent?The time it takes for the ball to reach its maximum height.

AThe maximum height reached by the ball.

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

CThe starting height of the ball.

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

AThe maximum height reached by the ball.

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

CThe starting height of the ball.

DFor what $x$

`x`-values does this model make sense?From $x=0$

`x`=0 to $x=4$`x`=4.AFrom $x=0$

`x`=0 to $x=20$`x`=20.BAll $x$

`x`values greater than $0$0.CFrom $x=0$

`x`=0 to $x=4$`x`=4.AFrom $x=0$

`x`=0 to $x=20$`x`=20.BAll $x$

`x`values greater than $0$0.C

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ^Equations using all available types of expressions, including simple root functions

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. '''Include rational, square root and cube root; emphasize selection of appropriate models.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. '''Include rational, square root and cube root; emphasize selection of appropriate models.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. '''Include rational and radical; Focus on using key features to guide selection of appropriate type of model function