Exam 12: Mathematical Problem Set

The graph of $y=-3+a^{2-x}$ for a>1 is best represented by which graph?

Graph the relation defined by the parametric equations when $-2 \leq t \leq 3$ $x(t)=t^{2}-5, y(t)=t-1$

Solve the system by any method. 3 x+4 y-6 z=4 x+2 y=8 5 x+2z=4

Solve the inequality $|x-5| \geq 3$ in interval notation.

For the function y=4+ln (x+1) , what is the inverse function, $f^{-1} ?$

Determine the range of $f(x)=13-20 x-x^{2}-3 x^{4}$ Express your answer in interval notation.

Find the equation that describes the set of all points (x, y, z) that are 3 units from the point (1,-2,6) .

The following three transformations are applied (in the given order) to the graph of $y=x^{2}:$ I. A vertical stretch by a factor of 0.3 II. A horizontal shift left 5 units III. A vertical shift up 6 units Write an equation for the graph produced as a result of applying these transformations.

Describe transformations that can be used to transform the graph of y=ln x to a graph of 5-ln (x-4) .

The matrix $\left[\begin{array}{rrr}1 / 2 & -1 & 1 / 2 \\1 / 2 & 0 & -1 / 2 \\-1 / 2 & 1 & 1 / 2\end{array}\right]$ is the inverse of $\left[\begin{array}{lll}1 & 2 & 1 \\0 & 1 & 1 \\1 & 0 & 1\end{array}\right]$ Solve the following system by the inverse matrix method. x+2 y+z =10 y+z =6 x+z =-8

At $3.5$ feet above home plate, a baseball is hit with the initial velocity of $125 \mathrm{ft} / \mathrm{sec}$ at an angle of elevation of $35^{\circ}$ . Let $t=0$ be the time when the ball is hit. Using the position function in parametric form, find the horizontal distance the ball will travel if air resistance is neglected.

Find the equation of the parabola that passes through (-6,22) and has vertex (2,-5) . Express your answer in the standard form for a quadratic function.

For the function $f(x)=\frac{-2}{x},$ what is the equation of the tangent line at x=2 ?

Write a proof by mathematical induction of the statement: $1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$

For the next 15 years, a small company's business volume can be modeled by the function $f(x)=47(1.09)^{x}+3 \sin \frac{\pi x}{6}$ where x is the number of years after 2006 and f is the sales in millions of dollars. (a) What are the company's sales in 2006? (b) What does the model project for sales in 2013? (c) How many years are in each economic cycle for this company?

Solve the double inequality $-8 \leq \frac{5 x+4}{2}<7$

Describe transformations that can be used to transform the graph of $\log x$ to a graph of $f(x)=4 \log (x+2)-3$

An airplane is flying on a bearing $75^{\circ}$ east of north at 600 mph. Express the velocity of the airplane as a vector.

When a car's brakes are applied, the speed of the car is given by $v(t)=80-22 t$ (in feet per second). Use the area under the function to determine how many feet the car travels between the time the brakes are applied (t=0) and the time the car stops.

Find two complex number roots for the equation $x^{2}-6 x+58=0$