Algebra 1

Characteristics of a linear function

1. Has a single output for a given input. 2. The same input cannot produce a different output on different occasions. 3. The graph is a line, but may not be a straight vertical line (would represent #2). 4. A straight line through the graph of a linear function may not intersect the graph in more than one point. 5. A linear function represents a constant rate of change (hence a line) 6. Can be expressed in the form y = mx + b

Vertical line test

A graph represents a function when no vertical line passes through more than one point on the graph.

linear equation in two variables vs. linear function

A linear equation can be written in the form y = mx + b, where m and b are constants. The graph of a linear equation is a line. A linear function is an equation in two variables whose graph is a non vertical line. A linear function has a constant rate of change.

What is quadratic function?

A quadratic function is a nonlinear function that can be written in the standard form y = ax2 + bx + c, where a ≠ 0.

Relation

A relation pairs inputs with outputs. When a relation is given as ordered pairs, the x coordinate is the input and y coordinate the output.

Function

A relation that pairs each input with exactly one output. Two different inputs can have the same output, but an input must have exactly one output in order to be considered a function.

domain of a function

A set of all possible input values

Range of a function or relation

A set of all possible output values

Discrete vs. Continuous domain

A set of input values that only consist of certain numbers in an interval. A set of input values that consist of all numbers in an interval. In order to know whether the domain is discrete or continuous, you need to know the context of the function, not just the equation. all equations are continuous, but not all values may be meaningful. E.g. number of tickets as a function of cost of each ticket is a discrete set. I.e. you can't buy partial tickets.

Constant function

Graph is a horizontal line. y = mx + b y = b when x is 0

Is y = x^2 + 6 a linear function?

No, it is not. We cannot rewrite it in the form y = mx + b.

When are two non-vertical lines perpendicular?

Nonvertical lines are perpendicular if and only if their slopes are negative reciprocals. Horizontal and vertical lines are always perpendicular to each other.

Slope of a non vertical line

Rise over run

What is y intercept?

The value of y when x = 0

All linear functions are linear equations. True or false?

True. Converse is false. Equation that makes a straight vertical line is not a function.

When are two non-vertical lines parallel? What about vertical lines?

Two non-vertical lines are parallel when they have the same slope. Vertical lines are always parallel since they have the same slope (slope is 0).

How many points do you need on a linear equation graph to determine the solution set? Why?

Two. Rate of change is constant. Generally get the x and y intercept.

What is Y intercept in slope intercept form, y = mx + b

When x is 0, Y = b. b is the Y intercept.

Is y = 6 a linear function? Explain.

Yes it is. We can rewrite it in the form y = mx + b as follows y = 0x + 6

x-intercept

the x-coordinate of a point where a graph crosses the x-axis

slope intercept form

y = mx + b Where m is the slope

Point-slope form

y2 - y1 = m(x2 - x1)