Key Features of Graphs

What are the slope and the y-intercept of y=-5x - 3?

-5 is the slope and -3 is the y-intercept.

What are the slope and the y-intercept of y=x+7.5?

1 is the slope and 7.5 is the y-intercept.

What are the slope and the y-intercept of y=2x+4?

2 is the slope and 4 is the y-intercept.

Undefined

4/0 = undefined. Division by zero is undefined.

Rational number

A number that can be written as a ratio in the form a/b, where b does not equal zero

Irrational number

A number that can not be written as a ratio in the form a/b

Co-efficient

A number that is multiplied by the variable in an algebraic expression or equation.

Real numbers

All rational and irrational numbers

Standard form

Ax + By = C Example: 5x + 2y = 15

Non-linear

Does not graph as a line Examples: y=x^2, y=2^x

Linear equation

Equation whose solutions form a straight line on a coordinate plane.

Even functions

Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. Example of an even function: y = x squared.

Odd functions

Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. Examples of odd functions are x and x cubed

Linear

Graphs as a line Examples: y=2, x =2, y=x, y=-x, y=2x, y=-2x

Which quadratic graphs have parabolas opening downwards?

Graphs of negative x-squared equations. Lead co-efficient is always negative. Parabolas open down and look like a frown.

Which quadratic graphs have parabolas opening upwards?

Graphs of positive x-squared equations. Lead co-efficient is always positive. Parabolas open up and look like a smile.

How do greater and lesser co-efficient values of x-squared effect the shape of a parabolic curve?

Greater values squash the curve (inwards to y-axis) Lesser values expand it (away from y-axis)

Periodicity

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.

Slope-intercept form

Linear equation written in the form y = mx + b, where "m" is slope and "b" is the y-intercept.

Equation

Mathematical sentence showing that two expressions are equal.

Slope

Measure of the steepness of a line on a graph; the rise (change in y) divided by the run (change in x)

The 7 P's

Prior proper preparation prevents particularly poor performance.

Interval of decrease

Range which there is a decrease between 2 points on a graph.

Interval of increase

Range which there is an increase between 2 points on a graph.

Domain

Set of all first coordinates or x-inputs. Example set: {(2,3), (4,5), (6,7)} {2, 4, 6}

Range

Set of all second coordinates or y-outputs. Example set: {(2,3), (4,5), (6,7)} {3, 5, 7}

End behavior

The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.

All real numbers

The entire number line. (-∞ ,∞) or {-∞ < x < ∞}

Absolute maximum

The greatest value in a relationship.

Absolute minimum

The least value in a relationship.

Subsets of the real numbers

The real numbers include all the rational numbers, such as the integer −5, the fraction 4/3, all the irrational numbers such as √2 (1.41421356...) and π (3.14159265...)

What do the graphs of quadratics (x-squared) equations look like?

They are called parabolas. These "U" shaped graphs will be continuous and smooth. Some open upwards and some open downwards.

Constant

Value that does not change.

Symmetry

When you're graphing quadratics, you may need to locate the parabola's axis of symmetry. This imaginary line goes through the vertex of the parabola and divides the graph into two equal parts. (Like a butterfly's mid line)

Intercept

Where a graph crosses an axis.

Axis of symmetry formula

x = -b/2a

x-intercept

x coordinate of a point where a graph crosses the x-axis. (y coordinate of this point is zero)

What are the x and y intercepts of 3x - 4y = 9?

x intercept = (3, 0) y intercept = (0, -9/4) or (0, -2.25)

What are the x and y intercepts of 2x + 4y = 8?

x intercept = (4, 0) y intercept = (0, 2)

What are the x and y intercepts of x - 3y = 9?

x intercept = (9, 0) y intercept = (0, -3)

y = 0 is the same as the _________.

x-axis

Point-slope form

y - y1 = m(x - x1) Use this formula, when we know: one point on the line and the slope of the line. y1 is "first y," x1 is the "first x"

y-intercept

y coordinate of a point where a graph crosses the y-axis (x coordinate of this point is zero)

x = 0 is the same as the _________.

y-axis

Integers

{..., -2, -1, 0, 1, 2, ...} All whole number and their opposites including zero

Whole numbers

{0, 1, 2, 3, ...}

Natural numbers (Also called counting numbers)

{1, 2, 3, ...}

Imaginary numbers

√-1, √-2, √-3, √-n, ... Imaginary numbers (√-1) are not real numbers.