4. Polynomial Functions

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

Height of an Object If an object is projected upward from an initial height of 100 ft with an initial velocity of 64 ft per sec, then its height in feet after t seconds is given by $s\left(t\right)=-16t^2+64t+100$. Find the number of seconds it will take the object to reach its maximum height. What is this maximum height?

Find a value of c so that y = x² - 10x + c has exactly one x-intercept.

For what values of a does y = ax² - 8x + 4 have no x-intercepts?

Define the quadratic function ƒ having x-intercepts (2, 0) and (5, 0) and y-intercept (0, 5).

Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept (0, 4).

The distance between the two points $P(x_1,y_1)$ and $R(x_2,y_2)$ is $d(P, R) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. Distance formula. Find the closest point on the line $y=2x$ to the point $(1,7)$. (Hint: Every point on $y=2x$ has the form $(x,2x)$, and the closest point has the minimum distance.)

A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution of the equation?

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes through the point (1,4).

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)²