Precalculus Chapter 2 Flashcards

Questions and Answers

What is the vertex of the graph of $-5(x + 4)^2 - 5$?

(5, 9)

(0, -15)

(-4, -5) (correct)

(-3, 4)

Identify the vertex of the parabola $F(x) = -3(x - 5)^2 + 9.

(5, 9)

Identify the vertex of the parabola $F(x) = 3x^2 + 18x + 31.$

(-3, 4)

Give the interval where the function $F(x) = (x + 5)^2 + 6$ increases.

[-5, infinity) Signup and view all the answers

Find the y-intercepts and any x-intercepts of the equation $y = x^2 - 2x - 15.$

Y = (0, -15), x = (5, 0), (-3, 0) Signup and view all the answers

Solve the quadratic inequality by graphing $x^2 - 4x + 3 \geq 0.$

(-infinity, 1] U [3, infinity) Signup and view all the answers

Factor the polynomial $x^3 + 5x^2 - 48x - 252$ completely given that $x - 7$ is a factor.

(x - 7)(x + 6)(x + 6) Signup and view all the answers

State the degree of the polynomial equation $-6x^2(x - 7)(x + 3)^3 = 0$.

Degree = 6 Signup and view all the answers

Find a polynomial equation with real coefficients that has the given roots: 2, -8, 3 + 5i.

(x - 2)(x + 8)(x - (3 - 5i))(x - (3 + 5i)) Signup and view all the answers

Use the rational zero theorem, Descartes's rule of signs, and the theorem on bounds as aids in finding all real and imaginary roots to the equation $x^4 + 15x^3 + 49x^2 - 15x - 50 = 0$.

-1, 1, -10, -5 Signup and view all the answers

Find all real solutions to the equation $\sqrt{x + 13} = x - 7$.

12 Signup and view all the answers

Find all real and imaginary solutions to the equation $(2m - 1)^2 - 4(2m - 1) - 21 = 0$.

-2, 3 Signup and view all the answers

Solve the absolute value equation $|x^2 - 10| = 4$.

-\sqrt{6}, \sqrt{6}, -\sqrt{14}, \sqrt{14} Signup and view all the answers

Describe the behavior of the function's graph at its x-intercepts for $f(x) = (x - 2)^2(x + 6)$.

(2, 0) touches, (-6, 0) crosses Signup and view all the answers

Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to \infty$ for $y = -x^5 - 2x^3 - 7x + 4$.

Y \to -\infty Signup and view all the answers

Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to -\infty$ for $y = 2x^4 + 2x^2 + x - 5$.

Y \to \infty Signup and view all the answers

Sketch the graph of the polynomial function $f(x) = -2x(x - 2)(x + 1)$.

(0, 0) crosses, (2, 0) crosses, (-1, 0) crosses Signup and view all the answers

Sketch the graph of the polynomial function $F(x) = (3x + 2)(x - 2)^2$.

(-2/3, 0) crosses, (2, 0) touches Signup and view all the answers

State the domain of the rational function $F(x) = \frac{16}{14 - x}$.

(-infinity, 14) U (14, infinity) Signup and view all the answers

Determine the domain and the equations of the asymptotes for the graph of the rational function given x-intercepts: (3, 0) and y-intercept: (0, 4).

Domain: (-infinity, 3) U (3, infinity), x = 3, y = 4 Signup and view all the answers

For the function $\frac{(x - 4)(x + 8)}{x^2 - 1}$, find all vertical asymptotes (if any).

x = 1, -1 Signup and view all the answers

For the function $\frac{6x^2 - 5x - 3}{5x^2 - 9x + 4}$, find all horizontal asymptotes (if any).

y = 6/5 Signup and view all the answers

Sketch the graph of the function $f(x) = \frac{2x + 1}{x}$. What are the vertical and horizontal asymptotes?

VA = 0, HA = 2 (check graph for curves) Signup and view all the answers

Study Notes

Quadratic Functions

The graph of ( -5(x+4)^2-5 ) opens downward with a vertex at (-4, -5).
The vertex of the parabola ( F(x) = -3(x-5)^2 + 9 ) is located at (5, 9).
For ( F(x) = 3x^2 + 18x + 31 ), the vertex is (-3, 4).

Function Behavior

The function ( F(x) = (x+5)^2 + 6 ) increases on the interval [-5, infinity).
The y-intercept of ( y = x^2 - 2x - 15 ) is (0, -15) and the x-intercepts are (5, 0) and (-3, 0).
The quadratic inequality ( x^2 - 4x + 3 \geq 0 ) has solutions in the intervals (-∞, 1] and [3, ∞).

Polynomial Operations

Factoring the polynomial ( x^3 + 5x^2 - 48x - 252 ) given the factor ( x - 7 ) results in ( (x-7)(x+6)^2 ).
The degree of the polynomial equation ( -6x^2(x-7)(x+3)^3 = 0 ) is 6.

Polynomial Roots

A polynomial with real coefficients having roots 2, -8, and ( 3+5i ) is expressed as ( (x-2)(x+8)(x-3-5i)(x-3+5i) ).

Solving Equations

The real solution for ( \sqrt{x} + 13 = x - 7 ) is ( x = 12 ).
The equation ( (2m-1)^2 - 4(2m-1) - 21 = 0 ) has real and imaginary solutions ( m = -2 ) and ( m = 3 ).

Absolute Value and Intercepts

The absolute value equation ( |x^2 - 10| = 4 ) yields solutions of ( -\sqrt{6}, \sqrt{6}, -\sqrt{14}, \sqrt{14} ).
The function ( f(x) = (x-2)^2(x+6) \ exhibits behavior at x-intercepts: (2, 0) touches and (-6, 0) crosses.

Leading Coefficient Test

For ( y = -x^5 - 2x^3 - 7x + 4 ), as ( x \to \infty ), ( y \to -\infty ).
The function ( 2x^4 + 2x^2 + x - 5 ) has ( y \to \infty ) as ( x \to -\infty ).

Graph Sketching

Sketching ( f(x) = -2x(x-2)(x+1) \ results in crossing points at (0,0), (2,0), and (-1,0).
The graph of ( F(x) = (3x + 2)(x - 2)^2 \ shows crossing at ((-2/3, 0)) and touching at (2, 0).

Rational Functions

The domain of the rational function ( F(x) = \frac{16}{14-x} ) is (-∞, 14) ∪ (14, ∞).
For the rational function with intercepts (3, 0) and (0, 4), the domain is (-∞, 3) ∪ (3, ∞), with asymptotes at ( x=3 ) and ( y=4 ).
The function ( \frac{(x-4)(x+8)}{x^2-1} \ has vertical asymptotes at ( x = 1 ) and ( x = -1 ).
A horizontal asymptote at ( y = \frac{6}{5} \ is present for ( \frac{6x^2 - 5x - 3}{5x^2 - 9x + 4} ).

Additional Graph Features

The function ( f(x) = \frac{2x + 1}{x} ) indicates a vertical asymptote at ( x = 0 ) and a horizontal asymptote at ( y = 2 ).

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Test your understanding of parabolic functions and their vertices with these flashcards from Precalculus Chapter 2. Each card presents a function and challenges you to identify key properties such as the vertex and intervals of increase or decrease. Perfect for reinforcing your knowledge of quadratic functions!