Q: Given $$f(x) = -2x^3$ + 5x^2$ - x + 4$$, what is the end behavior of the graph as $$x \to \infty$$ and $$x \to -\infty$$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to \infty$$

Question 1

Which of the following is the correct vertex form of a quadratic function?

Accepted Answer

$$f(x) = a(x - h)^2$ + k$$

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 4x + 1$$, what are the coordinates of its vertex?

Accepted Answer

$$(1, -1)$$

Question 3

For the quadratic function $$f(x) = -x^2$ + 6x - 5$$, does the parabola open upwards or downwards, and is the vertex a maximum or minimum?

Accepted Answer

Opens downwards; vertex is a maximum

Question 4

Which of the following is NOT a polynomial function?

Accepted Answer

$$f(x) = x$$^{1/2}$$ + 2x$$

Question 5

What is the degree and leading coefficient of the polynomial $$f(x) = 4x^5$ - 3x^2$ + x - 7$$?

Accepted Answer

Degree: 5, Leading coefficient: 4

Question 6

Given $$f(x) = -2x^3$ + 5x^2$ - x + 4$$, what is the end behavior of the graph as $$x 	o \infty$$ and $$x 	o -\infty$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 7

A polynomial function $$f(x)$$ has zeros at $$x = -2 ($$multiplicity 2) and $$x = 3 ($$multiplicity 1). Which statement is true about the graph at these zeros?

Accepted Answer

At $$x = -2$$, the graph touches the $$x-$$axis; at $$x = 3$$, it crosses the $$x-$$axis

Question 8

What is the maximum number of turning points for the polynomial $$f(x) = x^6$ - 2x^4$ + x$$?

Accepted Answer

5

Question 9

If a quadratic function is given in standard form $$f(x) = ax^2$ + bx + c$$, what is the formula for the axis of symmetry?

Accepted Answer

$$x = -\frac{b}{2a}$$

Question 10

Which transformation occurs to the graph of $$f(x) = x^2$ when it is changed to f(x) = (x$$ - $$3)^2 + 2$?

Accepted Answer

Shift right 3 units, shift up 2 units

Question 11

Given $$f(x) = 2(x - 1)^2$ - 7$$, what is the minimum or maximum value and where does it occur?

Accepted Answer

Minimum at $$x = 1$$, value $$-7$$

Question 12

Which of the following polynomial functions has the same end behavior on both sides?

Accepted Answer

$$f(x) = x^4$ - 2x^2$ + 1$$

Question 13

A real-world scenario: The height $$h (in $$meters) of a ball thrown upward is modeled by $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

$$22.41$$ meters

Question 14

Which of the following is the correct process to convert $$f(x) = x^2$ + 6x + 7 to $$vertex form?

Accepted Answer

Complete the square: $$f(x) = (x + 3)^2$ - 2$$

Question 15

Given $$f(x) = x^3$ - 4x^2$ + 4x$$, find all zeros and their multiplicities.

Accepted Answer

$$x = 0 ($$multiplicity 1), $$x = 2 ($$multiplicity 2)

Question 16

For the polynomial $$f(x) = -x^4$ + 2x^2$ - 1$$, what is the end behavior as $$x 	o \infty$$ and $$x 	o -\infty$$?

Accepted Answer

Both $$f(x) 	o -\infty$$

Question 17

A polynomial $$f(x)$$ has degree 7. What is the maximum number of turning points it can have?

Accepted Answer

6

Question 18

Which of the following graphs could NOT represent $$f(x) = x^4$ + 1$$ based ONLY on the maximum number of turning points?

Accepted Answer

A graph with 5 turning points

Question 19

Given $$f(x) = (x - 2)^2$(x + 1)$$, which statement about the zeros is correct?

Accepted Answer

$$x = 2 is a $$zero of multiplicity 2; $$x = -1 is a $$zero of multiplicity 1

Precalculus Study Guide: Quadratic and Polynomial Functions

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