Algebra 2 Chapter 2 Review

Questions and Answers

Is the relation (-3, -1), (-1, 2), (0, -3), (1, 2), (3, 0) a function? Identify the domain and range.

The relation is a function. D={-3, -1, 0, 1, 3}, R={-3, -1, 0, 2}

Is the relation (-2, -1), (-1, 1), (0, -1), (1, 1), (2, 0), (3, -1), (3, 2) a function? Identify the domain and range.

Not a function, fails vertical line test at x=3. D={-2, -1, 0, 1, 2, 3}, R={-1, 0, 1, 2}

Is the function f(x)=5x-4 linear? Evaluate the function when x=2.

Linear; f(2)=6

Is the function h(x)=2x^2 linear? Evaluate the function when x=2.

Not linear; h(2)=8 Signup and view all the answers

Is the function p(x)=-x^3-3 linear? Evaluate the function when x=2.

Not linear; p(2)=5 Signup and view all the answers

Is the function g(x)=-6/x-7 linear? Evaluate the function when x=2.

Not linear; g(2)=-10 Signup and view all the answers

Is the function h(x)=x-2 linear? Evaluate the function when x=2.

Linear; h(x)=0 Signup and view all the answers

Is the function f(x)=-2x+4 linear? Evaluate the function when x=2.

Linear; f(x)=0 Signup and view all the answers

Is the function g(x)=(-1/2)x+5 linear? Evaluate the function when x=2.

Linear; g(x)=4 Signup and view all the answers

Is the function p(x)=3x-1 linear? Evaluate the function when x=2.

Linear; p(x)=5 Signup and view all the answers

What is the slope of the line through the points (5, 2) and (1, -2)? Is the line rising, falling, horizontal, or vertical?

m=1; rising Signup and view all the answers

What is the slope of the line through the points (0, -6) and (0, 2)? Is the line rising, falling, horizontal, or vertical?

m=undefined; vertical Signup and view all the answers

What is the slope of the line through the points (-6, 3) and (-5, 1)? Is the line rising, falling, horizontal, or vertical?

m=-2; falling Signup and view all the answers

What is the slope of the line through the points (-6, -2) and (-3, -2)? Is the line rising, falling, horizontal, or vertical?

m=0; horizontal Signup and view all the answers

What is the slope of the line through the points (-4, 1) and (2, -1)? Is the line rising, falling, horizontal, or vertical?

m=-1/6; falling Signup and view all the answers

What is the slope of the line through the points (3, -4) and (3, 4)? Is the line rising, falling, horizontal, or vertical?

m=undefined; vertical Signup and view all the answers

Which line is steeper between Line 1 through (-3, 7) and (4, 3) and Line 2 through (5, -3) and (4, -2)?

Line 1: m=4/7; Line 2: m=-1; Line 2 is steeper Signup and view all the answers

Which line is steeper between Line 1 through (-8, -5) and (2, 0) and Line 2 through (-4, 6) and (5, 9)?

Line 1: m=1/2; Line 2: m=1/3; Line 1 is steeper Signup and view all the answers

Write the equation of the line described by a slope of -2 and y-intercept of 8.

y=-2x+8 Signup and view all the answers

Write the equation of the line described by a slope of 6 and a y-intercept of -3.

y=6x-3 Signup and view all the answers

Write the equation of the line described by a slope of -3 through the point (1, -4).

y+4=-3(x-1) Signup and view all the answers

Write the equation of the line described by a slope of 1/4 through the point (-8, 0).

y+8=(1/4)x Signup and view all the answers

Write the equation of the line described by the point (2, -3) parallel to y=3x-11.

y+3=3(x-2) Signup and view all the answers

Write the equation of the line described by the point (0, 8) perpendicular to y=(1/3)x+5.

y=-3x+8 Signup and view all the answers

Study Notes

Functions and Relations

A relation is a function if each input (x-value) has a unique output (y-value).
Example: The relation {(-3, -1), (-1, 2), (0, -3), (1, 2), (3, 0)} is a function. Domain: {-3, -1, 0, 1, 3}. Range: {-3, -1, 0, 2}.
Example: The relation {(-2, -1), (-1, 1), (0, -1), (1, 1), (2, 0), (3, -1), (3, 2)} is not a function (fails vertical line test at x=3). Domain: {-2, -1, 0, 1, 2, 3}. Range: {-1, 0, 1, 2}.

Linear vs Non-linear Functions

A function is linear if it can be expressed in the form y = mx + b.
Example: f(x) = 5x - 4 is linear. Evaluated at x=2, f(2) = 6.
Example: h(x) = 2x^2 is not linear. Evaluated at x=2, h(2) = 8.
Other non-linear functions include p(x) = -x^3 - 3 (p(2) = 5) and g(x) = -6/x - 7 (g(2) = -10).
Linear functions also include h(x) = x - 2 (h(2) = 0) and f(x) = -2x + 4 (f(2) = 0).

Evaluating Functions

For linear functions:
- g(x) = (-1/2)x + 5, evaluated at x=2 gives g(2) = 4.
- p(x) = 3x - 1, evaluated at x=2 gives p(2) = 5.

Slope of a Line

Slope (m) is determined using the formula m = (y2 - y1) / (x2 - x1).
Example: For points (5, 2) and (1, -2), the slope is m = 1 (line is rising).
Example: For points (0, -6) and (0, 2), the slope is undefined (line is vertical).
Example: For points (-6, 3) and (-5, 1), the slope is m = -2 (line is falling).
A slope of 0 indicates a horizontal line, as seen with points (-6, -2) and (-3, -2).

Steepness of Lines

Line 1 through points (-3, 7) and (4, 3) has slope m = 4/7.
Line 2 through points (5, -3) and (4, -2) has slope m = -1.
Line 2 is steeper than Line 1.
Example of steepness: Line 1 through (-8, -5) and (2, 0) has a slope of m = 1/2, while Line 2 through (-4, 6) and (5, 9) has a slope of m = 1/3; Line 1 is steeper.

Writing Equations of Lines

Equation format for a line given slope (m) and y-intercept (b) is y = mx + b.
Given slope = -2 and y-intercept = 8, the equation is y = -2x + 8.
Given slope = 6 and y-intercept = -3, the equation is y = 6x - 3.
For a line with slope = -3 through point (1, -4), the point-slope form is y + 4 = -3(x - 1).
For slope = 1/4 through point (-8, 0), the equation is y + 8 = (1/4)x.
For a line through point (2, -3) parallel to y = 3x - 11, the equation is y + 3 = 3(x - 2).
For a line through (0, 8) perpendicular to y = (1/3)x + 5, the equation is y = -3x + 8.

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Review key concepts from Algebra 2 Chapter 2 with this set of flashcards. Test your understanding of functions, domains, and ranges while identifying whether given relations are functions. Strengthen your algebra skills with practical examples.