91Ó°ÊÓ

True or False The slope of a nonvertical line is the average rate of change of the linear function.

The price p (in dollars) and the quantity x sold of a certain product obey the demand equation$p=-\frac{1}{10}x+1000$ .

(a) Find a model that expresses the revenue R as a function of x.

(b) What is the revenue if 400 units are sold?

(c) What quantity x maximizes revenue? What is the maximum revenue?

(d) What price should the company charge to maximize revenue?

In Problems 9–14, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing.

$f\left(x\right)=-\frac{1}{2}{x}^2+2$

Can a quadratic function have a range of$(-∞,∞)$ ? Justify your answer.

True or False If the average rate of change of a linear function is $\frac{2}{3}$, then if $y$increases by $3$,$x$will increase by$2$.

A projectile is fired from a cliff $200ft$above the water at an inclination of $45°$to the horizontal, with a muzzle velocity of role="math" localid="1647357108345" $50ft/s$. The height h of the projectile above the water is modeled by

$h\left(x\right)=\frac{-32x^2}{{\left(50\right)}^2}+x+200$

where x is the horizontal distance of the projectile from the face of the cliff.

Part (a): At what horizontal distance from the face of the cliff is the height of the projectile a maximum?

Part (b): Find the maximum height of the projectile.

Part (c): At what horizontal distance from the face of the cliff will the projectile strike the water?

Part (d): Using a graphing utility, graph the function h,$0≤x≤200$.

Part (e): Use a graphing utility to verify the solutions found in parts (b) and (c).

Part (f): When the height of the projectile is $100ft$above the water, how far is it from the cliff?

Consider the function $f\left(x\right)=\frac{x}{x+4}$.

(a) Is the point $\left(1,\frac{1}{4}\right)$on the graph of function.

(b) if $x=-2$, what is $f\left(x\right)$. What point is on the graph of f ?

(c) If $f\left(x\right)=2$, what is x? What point is on the graph of f ?

A projectile is fired at an inclination of $45°$to the horizontal, with a muzzle velocity of $100ft/s$. The height h of the projectile is modeled by

$h\left(x\right)=\frac{-32x^2}{{\left(100\right)}^2}+x$

where x is the horizontal distance of the projectile from the firing point.

Part (a): At what horizontal distance from the firing point is the height of the projectile a maximum?

Part (b): Find the maximum height of the projectile.

Part (c): At what horizontal distance from the firing point will the projectile strike the ground?

Part (d) Using a graphing utility, graph the function h, $0≤x≤350$.

Part (e): Use a graphing utility to verify the results obtained in parts (b) and (c).

Part (f): When the height of the projectile is $50ft$above the ground, how far has it traveled horizontally?

In Problems 9–14, (a) graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing.

$f\left(x\right)=9x^2+6x+1$

If $f\left(x\right)=2x+3$, then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant