Algebra Class: Equations and Functions

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Questions and Answers

A line passes through the point $(5, 3)$ and has a slope of $-\frac{2}{5}$. Using the point-slope form, which equation represents this line?

$y - 3 = \frac{2}{5}(x - 5)$
$y + 5 = -\frac{2}{5}(x + 3)$
$y - 3 = -\frac{2}{5}(x - 5)$ (correct)
$y + 3 = -\frac{2}{5}(x + 5)$

Which of the following represents the slope and y-intercept of the line given by the equation $4x + 2y = 16$?

Slope: 4, y-intercept: 16
Slope: -4, y-intercept: 16
Slope: 2, y-intercept: 8
Slope: -2, y-intercept: 8 (correct)

What are the domain and range of the quadratic function $y = x^2 + 3$?

Domain: $(0, ∞)$, Range: $[3, ∞)$
Domain: $(-∞, ∞)$, Range: $[3, ∞)$ (correct)
Domain: $(-∞, ∞)$, Range: $(3, ∞)$
Domain: $(-\sqrt{3}, \sqrt{3})$, Range: $[0, ∞)$

Given $f(x) = 5x - 3$ and $g(x) = -2x + 1$, what is the expression for $f(g(x))$?

<p>$-10x - 2$ (D)</p> Signup and view all the answers

Flashcards

Finding the equation of a line

To find the equation of a line, we need a point it passes through and its slope. We use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

Slope and y-intercept of a line

The slope of a line represents its steepness, and the y-intercept is the point where the line crosses the y-axis. To find them, we rewrite the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce.

Graphing a Piecewise Function

A piecewise-defined function is a function defined by multiple sub-functions, each with its own specific domain. To graph it, we graph each sub-function within its corresponding domain.

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Composition of Functions

Composition of functions involves applying one function to the output of another function. To find g(f(x)), we substitute the entire expression of f(x) into the variable x of g(x).

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Study Notes

Equation of a Line

Find the equation of a line given a point and slope using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is the point and m is the slope.
Example: A line passes through (2, 2) with a slope of -3. The equation is y - 2 = -3(x - 2).

Slope and Y-intercept

Find the slope and y-intercept of a linear equation in the form Ax + By = C.
Example: For 2x + 5y = 19, the slope is -2/5 and the y-intercept is 19/5.

Domain and Range of a Function

Find the domain and range of a quadratic function.
Example: For y = x² - 5, the domain is all real numbers (-∞, ∞) and the range is y ≥ -5.

Piecewise Function

Graph a piecewise-defined function by plotting points according to the defined conditions.
Example: For f(x)= {x+1 if x ≤ 0, x if x > 0}, graph the sections separately for x ≤ 0 and x > 0.

Function Composition

Find the composition of two functions (g(f(x))).
Example: If f(x) = -6x + 4 and g(x) = 2x + 7, then g(f(x)) = -12x + 15.

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Description

Test your understanding of key algebraic concepts, including finding equations of lines, determining slopes and intercepts, and analyzing domains and ranges of functions. This quiz will also cover piecewise functions and function compositions through example problems.