Algebra Functions and Graphs

Play an AI-generated podcast conversation about this lesson

What is the primary characteristic of a function?

It passes through the origin (0,0) only.

It has at least one x-value that pairs with multiple y-values.

It has all ordered pairs in positive coordinates.

No x-value is paired with two different y-values. (correct)

In which quadrant would the point (-3, 5) be located?

Quadrant III

Quadrant II (correct)

Quadrant I

Quadrant IV

How do you determine the domain of the function f(x) = √(3x - 10)?

Solve 3x - 10 > 0.

Solve 3x - 10 ≥ 0. (correct)

Analyze the graph for intercepts.

Set the function equal to 0.

What should you do if the denominator of f(x) = 1/(x - 2) is equal to zero?

Exclude x = 2 from the domain. Signup and view all the answers

What is represented by f(x)?

A function's output. Signup and view all the answers

What is the slope of the line represented by the equation 3x + 4y = 12?

-3/4 Signup and view all the answers

How do you determine the y-intercept from the equation y = -3/4x + 3?

Set x = 0 Signup and view all the answers

In the context of a line's slope, what is the slope of a horizontal line?

0 Signup and view all the answers

When given a point P(-1, 2) and a slope m = -4, what is the equation of the line in slope-intercept form?

y = -4x - 2 Signup and view all the answers

What is the key step in transforming the equation of a line from standard form to slope-intercept form?

Isolate y Signup and view all the answers

What does it mean if a graph is increasing?

The graph goes up from left to right. Signup and view all the answers

When expressed in interval notation, which of the following is the correct representation of a domain ending at 5?

(-∞, 5) Signup and view all the answers

Which characteristic defines an even function?

All powers of x are even numbers. Signup and view all the answers

In the piecewise function, what expression represents values of x that are greater than or equal to 0?

f(x) = 3x - 1 Signup and view all the answers

What does it indicate when a graph is constant?

The values of the function do not change as x varies. Signup and view all the answers

What effect does a negative sign outside of f(x) have on the graph?

It flips the graph down. Signup and view all the answers

Which of the following manipulations will move the graph of f(x) to the right?

f(x + 2) Signup and view all the answers

What happens to the graph when a number between 0 and 1 is multiplied in front of f(x)?

The graph becomes shorter and fatter. Signup and view all the answers

What does a transformation of f(-x) mean for the graph?

The graph flips left to right. Signup and view all the answers

How does f(x) + 5 affect the graph of f(x)?

The graph moves up by 5. Signup and view all the answers

What is the domain of the function $f(x) = \frac{3x+1}{x-2}$?

(-\infty, 2) \cup (2, \infty) Signup and view all the answers

For the function $f(x) = \sqrt{3x - 12}$, what is the minimum value of x within its domain?

4 Signup and view all the answers

When composing the functions $f(x) = 3x - 5$ and $g(x) = 2x + 4$, what is the result of $fog(x)$?

$6x + 7$ Signup and view all the answers

What does the expression $gof(x)$ equal when $g(x) = 2x + 4$ and $f(x) = 3x - 5$?

$6x - 6$ Signup and view all the answers

Why must $x$ be excluded from the domain of a function that has a denominator?

Because it leads to division by zero Signup and view all the answers

What is the formula for calculating the distance between two points (x1, y1) and (x2, y2)?

Distance = $ ext{sqrt}((x_2-x_1)^2 + (y_2-y_1)^2)$ Signup and view all the answers

What is the standard equation of a circle with center (h, k) and radius r?

(x-h)² + (y-k)² = r² Signup and view all the answers

From the circle equation $(x+4)² + y² = 25$, what is the radius?

5 Signup and view all the answers

What is the midpoint of the points (2, 6) and (4, 8)?

(3, 7) Signup and view all the answers

What is the center of the circle represented by the equation $(x-2)² + (y+3)² = 4$?

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

What characteristic is true of parallel lines?

They have the same slope. Signup and view all the answers

What can be said about the slopes of perpendicular lines?

They are opposite and reciprocal. Signup and view all the answers

If one line has a slope of $2$, what is a possible slope for a line that is perpendicular to it?

$-1/2$ Signup and view all the answers

Which statement is false regarding parallel and perpendicular lines?

Parallel lines intersect each other. Signup and view all the answers

How do you identify two lines that are parallel?

By having the same slope. Signup and view all the answers

What defines an odd function?

All powers of x are odd. Signup and view all the answers

What is the symmetry property of odd functions?

Symmetry to the origin. Signup and view all the answers

Which of the following functions is not classified as odd?

2x^3 - 6x^2 + 4x - 5x^0 Signup and view all the answers

Which statement is true concerning constants in odd functions?

Constants are always even. Signup and view all the answers

What can be concluded about the function $x^5 + 2x^3 - x$?

It is odd. Signup and view all the answers