On a trip to St. Louis, you visit the Gateway Arch. Since you have plenty of time on your hands, you decide to estimate its altitude. You set up a Cartesian coordinate system with one end of the arch at the origin. The other end of the arch is at $x=162$ meters. To find a third point on the arch, you measure a value of $y=4.55$ meters when $x=1 \mathrm{me}-$ ter. You assume that the arch is parabolic.
a. Find the particular equation of the underside of the arch.
b. What is the $x$-coordinate of the vertex? By substituting this number into the equation, predict the height of the arch.
c. An airplane with a wingspan of 40 meters tries to fly through the arch at an altitude of 170 meters. Could the plane possibly make it? Justify your answer.

The Gateway Arch in St. Louis is often mistaken to be parabolic in shape. In fact, it is a catenary, which has a more complicated formula than a parabola. The Arch is 630 feet high and 630 feet wide at its base.

The table below gives the height of the Arch at various widths; find the corresponding heights for the parabola found in (a).

$$\begin{array}{|cc|} \hline \text { Width (ft) } & \text { Height (ft) } \\ \hline 567 & 100 \\ 478 & 312.5 \\ 308 & 525 \\ \hline \end{array}$$

The Gateway Arch in St. Louis is often mistaken to be parabolic in shape. In fact, it is a catenary, which has a more complicated formula than a parabola. The Arch is 630 feet high and 630 feet wide at its base.

Find the equation of a parabola with the same dimensions. Let x equal the horizontal distance from the center of the arch.

A construction company is building a new suspension bridge that has support cables attached to a center tower at various heights. One cable is attached at a height of 30 feet and connects to the ground 50 feet from the base of the tower. If the support cables should run parallel to each other, how far from the base should the company attach a cable whose other end is connected to the tower at a height of 25 feet?

What is the minimum number of data points you need to find a single quadratic model for a data set? Explain.

Dee Side has a pig that presently weighs 200 pounds. She could sell it now for a price of $\$ 1.40$ a pound.
a. What is the worth of the pig now?
b. The pig is gaining 5 pounds a week. Write an equation for its weight as a function of weeks. What kind of function is this?
c. The price per pound is dropping 2 cents a week. Write an equation for price per pound as a function of weeks. What kind of function is this?
d. Write an equation for the total worth of the pig as a function of weeks. What kind of function is this?
e. Predict the worth of the pig after 8 weeks, 16 weeks, and 24 weeks.
f. When should Dee sell the pig to get the maximum amount of money for it?
g. Sketch the graph of worth versus time in a reasonable domain.

A gardener is putting a wire fence along the edge of his garden to keep animals from eating his plants. If he has 20 meters of fence, what is the largest rectangular area he can enclose?

• To find the area of a rectangle, what two quantities do you need? Choose one to be your variable and write the other in terms of this variable.
• How can a graph help you solve this problem?
• What quadratic function represents the area of the garden?

A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters from the center. (Assume that the road is level.)

The Verrazano-Narrows Bridge in New York City is the longest suspension bridge in North America. The Golden Gate Bridge in San Francisco is $60$ feet shorter than the Verrazano-Narrows Bridge. If the length of the Verrazano-Narrows Bridge is $m$ feet, express the length of the Golden Gate Bridge as an algebraic expression in $m$.

The number of bushels of barley on an acre of land will yield depends on how many seeds per acre you plant. From previous planting statistics, you find that if you plant 2 million seeds per acre, you can harvest 22 bushels per acre, and if you plant 4 million seeds per acre you can harvest 40 bushels per acre. As you plant more seeds per acre, the harvest will reach a maximum, then decrease. This happens because the young plants crowd each other out and compete for food and sunlight. Assume, therefore, that the number of bushels per acre you can harvest varies quadratically with the number of millions of seeds per acre you plant.
a. Write three ordered pairs of (millions of seeds per acre, bushels per acre). The third ordered pair is not given above but should be obvious.
b. Write the particular equation for this function.
c. How many bushels per acre would you expect to get if you plant 16 million seeds per acre?
d. Based on your model, would it be possible to get a harvest of 70 bushels per acre? Justify your answer. e. How much should you plant to get the maximum number of bushels per acre?
f. According to your model, is it possible to plant so many seeds that you harvest no barley at all?
g. Plot the graph of this quadratic function in a suitable domain.

Plot the graph accurately by making a table of values of $x$ and $y$. You may use different scales on the two axes, if necessary, to make the graphs have reasonable proportions.
$y=-3 x^2+5 x-10$

Solve the equation .
$4 x^2-8 x+5=x+3$

Plot the complex number on a complex number plane.\

$$-6+i$$

Solve the equation .
$2 x^2+2 x-2=x-x^2$