Determining if Linear
Determine whether the function \( f(x) = 3x + 7 \) is linear.
Rewriting as an Equation
Given the function \(f(x) = x^2 + 3x - 4\), express \(f(x+2)\) as an equation.
Rewriting as y=mx+b
Rewrite the function \(f(x) = 2x^2 - 6x + 5\) in the form \(y = mx + b\).
Find the Behavior (Leading Coefficient Test)
Given the polynomial function \(f(x) = -2x^5 + 3x^4 - 7x^3 + 2x^2 - x + 1\), describe the end behavior of the function using the Leading Coefficient Test.
Finding Ordered Pair Solutions
Find the ordered pair solutions for the function \(f(x) = 3x^2 - 2x + 1\) when \(x = -1, 0, 1\).
Determining Odd and Even Functions
3. Write equations for two functions of even symmetry with 3 x-intercepts, two of which are x=1 and x=-1 (both multiplicity 1 ).
Finding the Symmetry
Given the function \( f(x) = -2x^3 + 3x^2 - 2x + 1 \), determine whether it is symmetrical about the y-axis (even), symmetrical about the origin (odd), or neither.
Finding the Asymptotes
Find the vertical and horizontal asymptotes of the function \( f(x) = \frac{2x^2 + 3x - 2}{x - 1} \).
Difference Quotient
Find the difference quotient for the function \( f(x) = 3x^2 - 2x + 1 \)
Finding Roots Using the Factor Theorem
Find the roots of the polynomial function \(f(x) = x^3 - 6x^2 + 9x - 4\) using the Factor Theorem.
Finding All Possible Roots/Zeros (RRT)
Find all possible roots of the function \(f(x) = 3x^3 - 2x^2 - 5x + 2\)
Finding Upper and Lower Bounds
Find the upper and lower bounds of the function \(f(x) = x^2 - 4x + 4\) on the interval [0, 3].
Finding the Vertex
Find the vertex of the function \( f(x) = 2x^2 - 5x + 3 \).
Rewriting as a Function
Graph the following function using transformation techniques. $g(x)=\frac{-2}{x-1}$
Determining if a Function is Rational
Sketch the graph of the following function. Indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur.
\[
f(x)=\frac{-3}{x-7}
\]
Determine the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has two horizontal asymptotes. The top asymptote is and the bottom asymptote is (Type equations.)
B. The function has one horizontal asymptote, $y=0$. (Type an equation.)
c. The function has no horizontal asymptotes.
Determine the slant asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The slant asymptote(s) is(are)
(Type an equation. Use a comma to separate answers as needed.)
B. The function has no slant asymptotes.
On what interval(s) is $\mathrm{f}$ concave up and on what interval(s) is $\mathrm{f}$ concave down? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function is concave up on and concave down on (Simplify your answers. Type your answers in interval notation. Use a comma tir separate answers as needed.)
B. The function is concave up on and is never concave down. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
C. The function is concave down on $\square$ and is never concave up. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
Determining if a Function is Proper or Improper
Determine if the function \( f(x) = \frac{5x^2 + 3x + 2}{x^2 + 4x + 4} \) is a proper or improper function.
Maximum/Minimum of Quadratic Functions
A coffee shop currently sells 480 lattes a day at $\$ 2.75$ each. They recently tried raising the by price by $\$ 0.25$ a latte, and found that they sold 40 less lattes a day.
a) Assume that the number of lattes they sell in a day, $N$, is linearly related to the sale price, $p$ (in dollars). Find an equation for $N$ as a function of $p$.
\[
N(p)=
\]
b) Revenue (the amount of money the store brings in before costs) can be found by multiplying the cost per cup times the number of cups sold. Again using $p$ as the sales price, use your equation from above to write an equation for the revenue, $R$ as a function of $p$.
\[
R(p)=
\]
c) The store wants to maximize their revenue (make as much money as possible). Find the value of $p$ that will maximize the revenue (round to the nearest cent).
\[
p=
\]
which will give a maximum revenue of $\$$
Finding the Slope
Find the slope of the line tangent to the function \(f(x) = 3x^2 - 2x + 1\) at the point where \(x = 2\).
Graphing
Graph the function \( f(x) = x^{3} - 6x^{2} + 9x + 15 \) and determine its x-intercepts.