Exam 3: Functions and Graphs

Solve the problem. -A winery can produce red wine, white wine, or a combination of the two. The product exchange function for red wine, $x$ , and white wine, $y$ , in tons is: $y=\frac{144,000-45 x}{1920+x}$ By sketching the quadrant I portion of the graph of this function, estimate the maximum quantity of each kind of wine that can be produced.

Solve the problem. -Suppose the price $p$ of bolts is related to the quantity $q$ that is demanded by: $p=540-5 q^{2}$ Where $q$ is measured in hundreds of bolts. Suppose the supply function for bolts is given by $\mathrm{p}=5 \mathrm{q}^{2}$ Where $\mathrm{q}$ is the number of bolts (in hundreds) that are supplied at price $\mathrm{p}$ . Find the equilibrium supply.

Graph the function. - $y=x^{2}-1$

Use a graphing calculator to find a viewing window that shows a complete graph of the given polynomial function(that is, a graph that includes all the peaks and valleys and indicates how the curve moves away frem the $x$ axis at thefar left and far right.) There are many possible correct answers. - $f(x)=x^{3}+4 x^{2}-5 x+4$

Solve the problem. -Suppose that the speed of a particle moving on the $x$ -axis at time $t$ seconds $(t \geq 0)$ is given by: $V(t)=t^{3}-8 t^{2}+15 t+10$ By sketching a graph of $C(t)$ , estimate at what time the speed of the particle is a minimum.

Determine whether the following rule defines $y$ as a function of $x$ . - $y=\frac{9 x}{5 x-2}$

Evaluate the function. -Given that $f(x)=-8$ , find $f(3.1)$

Graph the function. - $y=-|5-x|$

State the domain of the given function. - $y=\sqrt{15-x}$

Solve the problem. -The water is drained from a tank by opening a valve at the bottom of the tank. The depth of the water in the tank thours after the valve is opened is $d(t)=6\left(1-\frac{t}{12}\right)^{2}$ meters. If the valve is opened at $1 \mathrm{pm}$ , what is the depth of water at $10 \mathrm{pm}$ ?

Solve the problem. -A ball is thrown vertically upward at an initial speed of $80 \mathrm{ft} / \mathrm{sec}$ . Its height (in feet) after $t$ seconds is given by $h(t)=t(80-16 t)$ What is the maximum height attained by the ball?

Solve the problem. -Suppose the price $p$ of bolts is related to the quantity $q$ that is demanded by: $p=620-4 q^{2}$ Where $\mathrm{q}$ is measured in hundreds of bolts. Suppose the supply function for bolts is given by $\mathrm{p}=4 \mathrm{q}^{2}$ Where $\mathrm{q}$ is the number of bolts (in hundreds) that are supplied at price $\mathrm{p}$ . Find the equilibrium price.

Determine the vertex of the parabola. - $f(x)=3 x^{2}-24 x+51$

Graph the function. -A car rental company charges $\$ 25$ per day or fraction of a day to rent a station wagon. Graph the ordered pairs (days, cost).

Write the word or phrase that best completes each statement or answers the question -Use a graphing calculator to graph $f(x)=\frac{2 x^{3}-3 x^{2}+x+1}{x^{2}+x+1}$ . Does the graph appear to have a horizontal asymptote? If so, what is the equation of the horizontal asymptote? Does the graph appear to have a nonhorizontal straight line as an asymptote? If so, what is the equation of the line?

Solve the problem. -John owns a hotdog stand. He has found that his profit is represented by the equation $P=-x^{2}+62 x+75$ , with $P$ being profits and $x$ the number of hotdogs. How many hotdogs must he sell to earn the most profit?

Graph the function. -A typing service charges $\$ 4$ plus $\$ 4$ per page or fraction of a page. Graph the ordered pairs. (pages, cost).

State whether the parabola opens upward or downward. - $f(x)=3 x^{2}+4 x+8$

Solve the problem. -Given the following revenue and cost functions, find the maximum profit. (Recall that profit equals revenue minus cost.) $R(x)=78 x-2 x^{2} ; C(x)=22 x+105$

Solve the problem. -Janice is running a 8354 meter race. Assume that she runs at a constant speed of 309 meters per minute. Find the rule of the function $f(t)$ that gives her distance from the finishing line $t$ minutes after the start of the race.