Question 1

Given the quadratic function $$f(x) = 1 - (x-3)^2$, what is the vertex of the parabola?

Accepted Answer

$$(3, 1)$$

Question 2

For the function $$f(x) = 1 - (x-3)^2$, what is the range of f(x$$)$$?

Accepted Answer

$$(-\infty, 1]$$

Question 3

Which of the following functions is NOT a polynomial?

Accepted Answer

$$h(x) = 4x^2$ + 2x$$^{3/2}$$ + 3x$$

Question 4

Given $$f(x) = 4x^4$ - 3x^2$ + 2x - 5$$, use the Leading Coefficient Test to determine the end behavior of the graph.

Accepted Answer

The graph of $$f(x)$$ rises to the left and rises to the right.

Question 5

Consider the polynomial $$f(x) = (x-5)(x+3)^2$. Which statement about the graph at x = -3$ is true?

Accepted Answer

The graph touches the x-axis and turns around at $$x = -3.$$

Question 6

A polynomial $$f(x) = x^4$ - 2x^2$ - 4 is $$evaluated at $$x=1$$ and $$x=4. If$$ $$f(1)$$ and $$f(4)$$ have opposite signs, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one real zero between $$x=1$$ and $$x=4.$$

Question 7

Given $$f(x) = 3x^3$ + 2x^2$ - x - 3$$, what are the $$x-$$intercepts and their multiplicities?

Accepted Answer

$$x = -3 ($$odd), $$x = 1 ($$odd)$

Question 8

For the function $$f(x) = -x^4$ + x^2$, what is the end behavior of the graph?

Accepted Answer

The graph falls to the left and falls to the right.

Question 9

Divide $$6x^3$$ - $$8x^2 + x + 1$ by x-2$ using long division. What is the remainder?

Accepted Answer

$$15$$

Question 10

Given $$x^3$$ + $$2x^2 - 5x - 6$ divided by x-2$ using synthetic division, what is the solution set for x^3$ + 2x^2$ - 5x - 6 = 0$$?

Accepted Answer

$$\{-3, -1, 2\}$$

Question 11

Which of the following is the set of all possible rational zeros for $$f(x) = x^3$ + 2x^2$ - 5x - 6$$?

Accepted Answer

$$\{\pm1, \pm2, \pm3, \pm6\}$$

Question 12

If a polynomial $$f(x)$$ has degree 3 and a positive leading coefficient, what is the end behavior of its graph?

Accepted Answer

Falls to the left and rises to the right.

Question 13

For the function $$f(x) = -x^4$ + x^2$, which of the following statements is true about its symmetry?

Accepted Answer

It is an even function, symmetric with respect to the $$y-$$axis.

Question 14

Given $$f(x) = x^3$ + 2x^2$ - 5x - 6$$, what is the $$y-$$intercept of the graph?

Accepted Answer

$$-6$$

Question 15

Which of the following best describes the behavior of the graph of $$f(x) = (x-2)^2$(x+1) at$$ $$x=2$$?

Accepted Answer

The graph touches the $$x-$$axis and turns around at $$x=2.$$

Question 16

If $$f(x) = x^3$ + 2x^2$ - 5x - 6$$ and you use synthetic division by $$x-2$$, what is the quotient?

Accepted Answer

$$x^2 + 4x + 3$

Question 17

Which of the following is the correct domain for the function $$f(x) = 1 - (x-3)^2$?

Accepted Answer

$$(-\infty, \infty)$$

Question 18

A real-world scenario: The height $$h (in $$meters) of a ball thrown upward is modeled by $$h(t) = -5t^2$ + 20t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

$$22$$ meters

Polynomial and Rational Functions: Study Notes and Worked Examples

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