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.