Chapter 1 Notes: Parent Functions, Transformations, and Linear Equations

A set of practice flashcards covering parent functions, their transformations, solving systems, and key forms for linear equations.

What is a parent function?

A basic function in a family; other functions are created by transforming the parent.

What is a transformation?

Shifts, dilations, or changes in a graph's orientation.

What is the constant function and its graph, domain, and range?

f(x) = c; a horizontal line; domain is all real numbers (R); range is {c}.

What is the linear parent function and its domain and range?

f(x) = x (a line with positive slope). Domain: R; Range: R.

What is the absolute value parent function and its domain and range?

f(x) = |x|; domain: R; range: [0, ∞).

What is the quadratic parent function and its domain and range?

f(x) = x^2; domain: R; range: [0, ∞).

What is an ordered triple in a system of three equations?

A set (x, y, z) that satisfies all three equations.

How do you typically solve a system of three equations in three variables?

Use substitution, elimination, or matrix methods to find the triple (x, y, z).

What is the slope-intercept form and its components?

y = mx + b; m is the slope, b is the y-intercept.

What does the slope m represent in y = mx + b?

The slope; rise over run.

What does the y-intercept b represent in y = mx + b?

The value of y when x = 0 (the point where the line crosses the y-axis).

What is the point-slope form and its components?

y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

What is a line of fit?

A line that best represents data points on a scatter plot and is used for estimation.

How do you estimate y when x is given using a line of fit?

Substitute the given x into the line's equation and compute y.

How does horizontal translation affect f(x)?

A horizontal shift is f(x - h) which moves the graph to the right by h; f(x + h) moves it left by h.

How does vertical translation affect f(x)?

A vertical shift is f(x) ± k, moving the graph up by k or down by k.

What happens when you reflect a function across the x-axis?

Replace f(x) with -f(x); the graph is flipped over the x-axis.

What happens when you reflect a function across the y-axis?

Replace x with -x to get f(-x); the graph is flipped over the y-axis.

What is a horizontal stretch or shrink?

In f(a x): 0 < a < 1 stretches (widens) horizontally; a > 1 shrinks (compresses) horizontally.

What is a vertical stretch or shrink?

In a f(x) multiplied by a: a > 1 stretches vertically; 0 < a < 1 shrinks vertically.