Exam 1: Functions and Graphs

Use an equation to solve the problem. -A square of side $x$ inches is cut out of each corner of an 8 in. by 12 in. piece of cardboard, and the sides folded up to form an open-topped box. Use your graphing calculator to determine the dimensions of cut-out squares that will produce the box of maximum volume. (Hint: you will first need to write the volume of the box $\mathrm { V }$ as a function of $x$ .)

Solve the problem. -The following information pertains to a bakery which makes donuts. Make a scatterplot of the data. Then graph the following four functions on the same coordinate system: $f _ { 1 } ( x ) =$ ? $f _ { 2 } ( x ) = 40 x + 1000 ; f _ { 3 } ( x ) = - x ^ { 2 } + 100 x ; f _ { 4 } ( x ) = - x ^ { 2 } + 80 x + 200$ . Which function best models the profit for $x$ cases of donuts?

Solve the problem. -The graph shows the relationship between the area A of a rectangle and the length $L$ , if the width is fixed. Find the area if the length is $20 \mathrm {~cm}$ .

Describe how to transform the graph of f into the graph of g. - $f ( x ) = ( x + 7 ) ^ { 2 } \text { and } g ( x ) = - ( x - 1 ) ^ { 2 }$

Match the function with the graph. -

Find two functions defined implicitly by the given relation. - $x + y ^ { 2 } = 64$

Find two functions defined implicitly by the given relation. - $x ^ { 2 } + y ^ { 2 } = 81$

Choose the one alternative that best completes the statement or answers the question. Graph the piecewise-defined function. - $h ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \text { if } x < 0 \\\sqrt { x } \text { if } x \geq 0\end{array} \right.$

Solve the problem. -The following information pertains to a bakery which makes donuts. Make a scatterplot of the data. Then graph the following two functions on the same coordinate system: $\mathrm { f } _ { 1 } ( \mathrm { x } ) = - \mathrm { x } ^ { 2 } + 100 \mathrm { x } ; \mathrm { f } _ { 2 } ( \mathrm { x } ) = 85 \mathrm { x }$ . Decide which function best models the data, and then use that function to estimate the profit associated with making 45 cases of donuts.

Match the numerical model to the corresponding model. - x 2 9 16 23 30 37 y 1 2 3 4 5 6

Find the domain of the given function. - $f ( x ) = \sqrt { x ^ { 4 } - 81 x ^ { 2 } }$

Choose the one alternative that best completes the statement or answers the question. Graph the piecewise-defined function. - $f ( x ) = \left\{ \begin{array} { l l } | x | & \text { if } x < 0 \\\cos x & \text { if } x \geq 0\end{array} \right.$

Find the asymptote(s) of the given function. - $h ( x ) = \frac { ( x - 5 ) ( x + 6 ) } { x ^ { 2 } - 4 }$ vertical asymptotes(s)

Use an equation to solve the problem. -When a number is added to its double and its triple, the sum is 138 . Find the three numbers.

Solve the problem. -The following data set gives the average home value, in dollars, for a city at 5 -year intervals. Determine where $f$ is increasing or decreasing.

Solve the problem. -In a chemistry class, 7 liters of a 4% silver iodide solution must be mixed with a 10% solution to get a 6% solution. How many liters of the 10% solution are needed?

Perform the requested operation or operations. - $f ( x ) = 4 x + 9 ; g ( x ) = 2 x - 1$ , find $f ( g ( x ) )$ .

Choose the one alternative that best completes the statement or answers the question. -What symmetry does the graph of $y = f ( x )$ exhibit?

Choose the one alternative that best completes the statement or answers the question. Solve the problem. - $y = ( x + 16 ) ^ { 2 } + 17$

Match the numerical model to the corresponding model. - x 135 7 9 11 y 2 0 6 20 4272