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Questions and Answers
What is the vertex of the graph of $-5(x + 4)^2 - 5$?
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.
Identify the vertex of the parabola $F(x) = -3(x - 5)^2 + 9.
(5, 9)
Identify the vertex of the parabola $F(x) = 3x^2 + 18x + 31.$
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.
Give the interval where the function $F(x) = (x + 5)^2 + 6$ increases.
Find the y-intercepts and any x-intercepts of the equation $y = x^2 - 2x - 15.$
Find the y-intercepts and any x-intercepts of the equation $y = x^2 - 2x - 15.$
Solve the quadratic inequality by graphing $x^2 - 4x + 3 \geq 0.$
Solve the quadratic inequality by graphing $x^2 - 4x + 3 \geq 0.$
Factor the polynomial $x^3 + 5x^2 - 48x - 252$ completely given that $x - 7$ is a factor.
Factor the polynomial $x^3 + 5x^2 - 48x - 252$ completely given that $x - 7$ is a factor.
State the degree of the polynomial equation $-6x^2(x - 7)(x + 3)^3 = 0$.
State the degree of the polynomial equation $-6x^2(x - 7)(x + 3)^3 = 0$.
Find a polynomial equation with real coefficients that has the given roots: 2, -8, 3 + 5i.
Find a polynomial equation with real coefficients that has the given roots: 2, -8, 3 + 5i.
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 $x^4 + 15x^3 + 49x^2 - 15x - 50 = 0$.
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 $x^4 + 15x^3 + 49x^2 - 15x - 50 = 0$.
Find all real solutions to the equation $\sqrt{x + 13} = x - 7$.
Find all real solutions to the equation $\sqrt{x + 13} = x - 7$.
Find all real and imaginary solutions to the equation $(2m - 1)^2 - 4(2m - 1) - 21 = 0$.
Find all real and imaginary solutions to the equation $(2m - 1)^2 - 4(2m - 1) - 21 = 0$.
Solve the absolute value equation $|x^2 - 10| = 4$.
Solve the absolute value equation $|x^2 - 10| = 4$.
Describe the behavior of the function's graph at its x-intercepts for $f(x) = (x - 2)^2(x + 6)$.
Describe the behavior of the function's graph at its x-intercepts for $f(x) = (x - 2)^2(x + 6)$.
Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to \infty$ for $y = -x^5 - 2x^3 - 7x + 4$.
Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to \infty$ for $y = -x^5 - 2x^3 - 7x + 4$.
Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to -\infty$ for $y = 2x^4 + 2x^2 + x - 5$.
Use the leading coefficient test to determine whether $y \to \infty$ or $y \to -\infty$ as $x \to -\infty$ for $y = 2x^4 + 2x^2 + x - 5$.
Sketch the graph of the polynomial function $f(x) = -2x(x - 2)(x + 1)$.
Sketch the graph of the polynomial function $f(x) = -2x(x - 2)(x + 1)$.
Sketch the graph of the polynomial function $F(x) = (3x + 2)(x - 2)^2$.
Sketch the graph of the polynomial function $F(x) = (3x + 2)(x - 2)^2$.
State the domain of the rational function $F(x) = \frac{16}{14 - x}$.
State the domain of the rational function $F(x) = \frac{16}{14 - x}$.
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).
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).
For the function $\frac{(x - 4)(x + 8)}{x^2 - 1}$, find all vertical asymptotes (if any).
For the function $\frac{(x - 4)(x + 8)}{x^2 - 1}$, find all vertical asymptotes (if any).
For the function $\frac{6x^2 - 5x - 3}{5x^2 - 9x + 4}$, find all horizontal asymptotes (if any).
For the function $\frac{6x^2 - 5x - 3}{5x^2 - 9x + 4}$, find all horizontal asymptotes (if any).
Sketch the graph of the function $f(x) = \frac{2x + 1}{x}$. What are the vertical and horizontal asymptotes?
Sketch the graph of the function $f(x) = \frac{2x + 1}{x}$. What are the vertical and horizontal asymptotes?
Flashcards
Vertex of a Parabola
Vertex of a Parabola
The highest or lowest point on a parabola.
Negative Leading Coefficient
Negative Leading Coefficient
The parabola opens downward.
Y-intercept
Y-intercept
The point where the graph crosses the y-axis.
X-intercepts
X-intercepts
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Quadratic Inequality Solution
Quadratic Inequality Solution
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Factoring Polynomials
Factoring Polynomials
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Degree of a Polynomial
Degree of a Polynomial
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Polynomial Root
Polynomial Root
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Complex Root
Complex Root
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Solving Radical Equations
Solving Radical Equations
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Extraneous Solution
Extraneous Solution
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Solving Equations by Substitution
Solving Equations by Substitution
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Absolute Value
Absolute Value
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Touching at x-intercept
Touching at x-intercept
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Crossing at x-intercept
Crossing at x-intercept
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Leading Coefficient Test
Leading Coefficient Test
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End Behavior (x→∞, y→-∞)
End Behavior (x→∞, y→-∞)
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End Behavior (x→-∞, y→∞)
End Behavior (x→-∞, y→∞)
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Graph Sketching
Graph Sketching
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Rational Function
Rational Function
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Domain of a Function
Domain of a Function
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Asymptote
Asymptote
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Vertical Asymptote
Vertical Asymptote
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Horizontal Asymptote
Horizontal Asymptote
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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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