Class Notes Functions And Linear Equations Final

Class Notes Functions and Linear Equations Linear Equations can be shown in five (5) ways. Given one of the five (5) ways, the other four (4) ways can be figured out. Linear Equation: y = mx + b y = 5x + 3 Function Table: x

y

0

3

1

8

2

13

Ordered Pairs:

(x, y) (0, 3) (1, 8) (2, 13)

A Set of Numbers in the Domain (x-values) and in the Range (y-values Graph:

= {0, 1, 2} = {3, 8, 13}

Linear Equations are equations that can be graphed on a coordinate plane (graph paper) with a straight line. The word line can be seen in the word linear. y-intercept form: One way to write a linear equation is using the y-intercept form: y = mx + b This linear equation format tells where the line will cross the y-axis.

In this equation, m and b will be numbers that are shown. Here is an example:

In this example,

y = mx + b y = 5x + 3 m = 5 and b = 3

m = slope m is always used to show the slope of a line on a graph. The m in this equation is a coefficient. Coefficients are numbers that are written smushed up against a variable. In the y-intercept format: y = mx + b the coefficient = m. In the example: y = 5x + 3 the coefficient = 5. A coefficient is a number that is not fully known, because it is yet to be multiplied by the variable; once the variable is known. 5x could be

5(3) if x = 3, which would become 15 or 5(21) if x = 21, which would become 105 or 5(-2) if x = -2, which would become -10

Note: IF the number is multiplied to a variable with a × sign in between the number and the variable, then the number is not called a coefficient. It is called a constant. A constant is a number that stands alone in an equation and is not attached directly to a variable. Example: 5 × 3 both the 5 and the 3 are constants.

b = y-intercept b is always used to show the y-intercept of a line. The y-intercept is the place on the graph where the line crosses the y-axis. NOTE: The value for x is ALWAYS 0, where the line crosses the y-axis at the y-intercept. So x = 0 at the y-intercept. In our example, the line will cross the y-axis, where y = 3. This can be determined from: The equation:

y = mx + b y = 5x + 3

The 3 is in the place of the b. This means the line crosses the y-axis where y =3.

The function table: x

y

0

3

1

8

2

13

Look at the row where x = 0, find the value for y. In this example, y = 3 where = 0 So, the straight line crosses the y-axis, where y = 3. The ordered pair:

Choose the ordered pair where x = 0. (x, y) (0,3) Then determine the value for y, when x = 0. y = 3. So, the straight line crosses the y-axis, where y = 3.

How to recognize a linear equation. Linear equations have two variables, most often shown as x and y. Both x and y are raised to the first power; x1 and y1 NOTE: Most often the exponent 1 is kept invisible, so what is seen is only x and y. In comparison, other types of equations have different powers of x. The x and y variables must be in separate terms (not multiplied together and separated from each other by a plus or minus sign) The most common written form of a linear equation is the y-intercept form. y = mx + b But they can also be written with the x and the y on the same side of the equal sign, like this: mx + y = b Each term (number and/or variable separated by a plus or minus sign) may be positive or negative.

There are other types of equations. Quadratic equation: y = ax2 + bx + c Notice that the x is squared or raised to the power of 2. A quadratic equation graphed:

or

Cubic equation: y = ax3 + bx2 + cx + d Notice that the first x is cubed or raised to the power of 3. A cubic equation graphed

Exponential equation: y = 2x Notice that now the x is the exponent. (The 2 is just used to illustrate (show) that a base number is raised to the power of x.) An exponential equation graphed.

Inverse variable: y=1 x Notice that now x is in the denominator of the fraction. This is the same as saying that y = x-1, (but we have not learned this math yet). The inverse variable equation graphed.