Algebra Functions and Equations Quiz

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Questions and Answers

What is the equivalent exponential expression for log_b 49 = 3?

b = 49^3

b^3 = 49 (correct)

b^49 = 3

b^3 = 3

What is the domain of the function f(x) = log(x + 4)?

x > -4 (correct)

x > -5

x < -4

x = -4

Which expression is equivalent to 4 log t - log s, expressed as a single logarithm?

log(t^4 + s)

lograc{t^4}{s} (correct)

log(t^4 * s)

log(t^4 - s)

What is the solution to the equation 2^x = 8?

x = 3 Signup and view all the answers

What is the solution to the system of equations 5x + 4y = 1 and 6x - 2y = -26?

(-3, 4) Signup and view all the answers

Which of the following equations represents a circle with center at (6, -9) and radius 12?

(x - 6)^2 + (y + 9)^2 = 144 Signup and view all the answers

For the polynomial $f(x) = 5(x - 6)(x + 1)^4$, what can be concluded about its behavior at the x-intercepts?

It crosses the x-axis at x = 6 and touches at x = -1. Signup and view all the answers

What is the value of y in the equation y = -8x + 28 when x = 3?

12 Signup and view all the answers

What does the equation y = -9x + 31 represent in terms of graph characteristics?

A line that decreases as x increases. Signup and view all the answers

Identify the potential rational zeros of the polynomial function $f(x) = 8x^4 + 31x^3 - 4x^2 + x - 4$.

$rac{1}{8}, -4, -1$ Signup and view all the answers

What type of asymptote does the function $g(x) = rac{x^2 - 36}{x + 2}$ have?

Horizontal asymptote at y = 1 Signup and view all the answers

What is the result of solving the logarithmic equation log4(x) = 2?

16 Signup and view all the answers

Which of the following describes the set {x| x < 7}?

It includes all real numbers less than or equal to 7. Signup and view all the answers

Which statement is true regarding the function $f(x) = x^3 - 3x^2 - 5x + 39$ given that an x-intercept is at x = -3?

It has three distinct real solutions. Signup and view all the answers

What is the equation for the horizontal asymptote of the function $h(x) = rac{-x^2 + 16}{x^2 + 49}$?

y = -1 Signup and view all the answers

What are the x-intercepts of the equation $x^2 + y - 36 = 0$?

(6, 0) Signup and view all the answers

Which of the following is the standard form of the equation of a circle with radius 10 and center at (1, -9)?

$(x - 1)^2 + (y + 9)^2 = 100$ Signup and view all the answers

What is the equation of the line in slope-intercept form that contains the points (-2, 2) and (7, -4)?

$y = -rac{2}{3}x + rac{2}{3}$ Signup and view all the answers

Determine the domain of the function $f(x) = \frac{1}{x - 3}$.

$(- ext{∞}, 3) \cup (3, ext{∞})$ Signup and view all the answers

Find the center (h, k) and radius r of the circle described by the equation $(x - 6)^2 + (y + 9)^2 = 144$.

Center: (6, -9), Radius: 12 Signup and view all the answers

What is the average rate of change of the function $f(x) = 4x^3 - 8x^2 - 1$ from $x = 1$ to $x = 5$?

23 Signup and view all the answers

What type of function is represented by $f(x) = -3x^4 - x^2$?

Even Signup and view all the answers

Which transformation describes the graph of $F(x) = f(x + 2) - 1$?

Shift left 2 units and down 1 unit Signup and view all the answers

Study Notes

Equation Intercepts

Find the x-intercepts by setting y = 0 in the equation and solving for x.
Find the y-intercepts by setting x = 0 in the equation and solving for y.

Equation of a Line

Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Points: Use two points to find the slope (m = (y₂ - y₁) / (x₂ - x₁)). Substitute the slope and one point into the equation to solve for b.

Equation of a Circle

Standard form: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Function Values

Substitute the given x-value into the function to find the corresponding y-value.

Function Domain

Determine the set of all possible input values (x-values) for which the function is defined.

Determining Even/Odd Functions

Even function: f(-x) = f(x)
Odd function: f(-x) = -f(x)

Intervals of Increase/Decrease/Constant

Increasing: As x increases, y increases.
Decreasing: As x increases, y decreases.
Constant: y stays the same as x increases.

Average Rate of Change

Find the change in y over the change in x between two points. (f(x₂)-f(x₁))/(x₂-x₁)

Composite Functions

Substitute the function g(x) into function f(x). (f(g(x)))

Inverse Functions

Switch x and y variables.
Solve for the new y.

Logarithms and Exponents

Convert between logarithmic and exponential forms.

Systems of Equations

Solve using matrix methods or other algebraic techniques.

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Description

Test your understanding of various algebraic concepts including finding intercepts, understanding the slope-intercept form, and working with functions. Dive into the definitions and characteristics of different functions like even and odd, and evaluate increasing or decreasing intervals. This quiz is perfect for reinforcing your knowledge in 10th-grade algebra.