Mom's Chapter 1

1.5

...

Intercept method of graphing an equation

1. Find the horizontal (x,0) intercept and vertical (0,y) intercept. 2. Plot the intercepts, and draw the line through the two points. Horizontal intercept is on the x axis, so the y = 0 (x,0) Vertical intercept is on the y axis, so the x = 0 (0,y)

Solution

A solution of an equation is a value of the variable that makes the equation true. When asked to find the solution, you solve the equation for the variable.

Independent variable

A variable (often denoted by x ) whose variation does not depend on that of another. Also called the input. When graphing this is your x axis.

dependent variable

A variable (often denoted by y ) whose value depends on that of another. Also called the output. When graphing this is your y axis.

Linear Inequality

An equation that is not equal. Using the less than (<), greater (>) signs. Also can be less than and equal to or greater than and equal to. When graphing an inequality on a line the point is open if it is less than or equal to and closed (filled in) if it is less than and equal to or greater than and equal to. On a graph the line is broken if the equation of an inequality is less than or greater than. The line is solid if the equation is less than or equal to; or greater than or equal to. REMEMBER: Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign.

Linear equation

An equation that when graphed forms a straight line.

General Form for a Linear Equation

Ax + By = C

Slope formula given 2 points on a graph Coordinate formula when 2 points on a line are given

Coordinate formula when 2 points on a line are given, is the slope formula

When a line has no y-intercept..... When a line has no x-intercept...

If a line has no y-intercept, that means it never intersects the y-axis, so it must be parallel to the y-axis. This means it is a vertical line, such as . This slope of this line is undefined. If the line has no x-intercept, then it never intersects the x-axis, so it must be parallel to the x-axis.

What is the definition of independent variable in math?

In mathematics, the "x" and "y" values in an equation or a graph are referred to as "variables." If an equation shows a relationship between x and y in which the value of y is dependent upon the value of x, y is known as the dependent variable and is sometimes referred to as 'function(x)' or f(x).

Linear Model

Linear models have equations of the form: y = (starting value) + (rate of change) ⋅ x

When getting mpg (miles per gallon) REMEMNER

Note that getting 12 miles per gallon is the same as using 1/12 gallon of gas per mile. Remember to have your units alike. Watch for minutes and hours, days and weeks, weeks and months, etc.

1.5 Slope-Intercept Form

Slope-Intercept Form We can write the equation of a line in the form y = mx + b where m is the slope of the line, and b is the y-intercept.

Know the difference between the slope formula and the point-slope formula.

The are the same formula used for different purposes. a. The slope formula is used to calculate the slope when we know two separate points and are looking for m (slope). m = ∆y / ∆x b. The point-slope formula is used to find the equation of a line. We know the m, and are looking for y = mx + b. review both formulas on previous cards.

y = mx + b If b (the y intercept) is 0 what happens if either x or y are 0?

The equation will pass through the origin.

point-slope form: y = y_1 + m(x − x_1)

The point-slope form is useful when we know the rate of change and one point on the line.

Intercepts

The points at which a graph crosses the axes are called the intercept of the graph. 1. To find the x-intercept, we set y = 0 and solve for x. 2. To find the y-intercept, we set x = 0 and solve for y.

slope-intercept form: y = b + mx

The slope-intercept form is useful when we know the initial value and the rate of change.

What is the difference between the slope formula m = y _2 − y_1 / x_2 − x_1 and the point-slope formula m = y − y_1 / x − x_1

They are really the same formula, but they are used for different purposes: a. The slope formula is used to calculate the slope when we know two points. We know (x_1, y_1) and (x_2, y_2), and we are looking for m. b. The point-slope formula is used to find the equation of a line. We know (x_1, y_1) and m, and we are looking for y = mx + b.

How do you find the equation of a line

Use the point-slope formula and solve for y, when given two points. m = ∆y / ∆x If you are given the slope and one point, first graph the point and count rise/run (the slope), to find the second point, graph the line. Now having the two points you use the point-slope formula to find the equation.

1.5 Slope-Intercept Form What we know.

What you learned: 1. The y-intercept of a line gives the initial value of y. 2. The slope of the line gives the rate of change of y with respect to x. Comparing these observations with the form for a linear model, we see that y = (starting value) + (rate) ⋅ x; which is---- y = b + mx Usually we write the terms in the opposite order, like this: y = mx + b. We call this last equation the slope-intercept form for a line.

Another form of Point slope formula Only one point given, use to find the equation of a line

When we use the slope formula to find the equation of a line, we substitute a variable point (x, y) for the second point.

Point slope formula Only one point given, use to find the equation of a line

When we use the slope formula to find the equation of a line, we substitute a variable point (x, y) for the second point.

Slope formula

m = ∆y / ∆x (Slope is the change in y over the change in X) y_2 − y_1 / x_2 − x_1 and x_2 ≠ x_1 Use this formula to find the slope of any two points on a line.

net change = final value − starting value To calculate the net change between two points on a number line, we can subtract their coordinates.

net change = final value − starting value For example, if you walk from 3rd street to 8th street, your distance, s, from the center of town has increased by 5 blocks, or ∆s = 8 − 3 = 5 If the temperature T drops from 28° to 22°, it has decreased by 6°, or ∆T = 22 − 28 = −6 The net change is positive if the variable increases, and negative if it decreases.

The Slope-intercept form, y=mx+b, is just a special case of the point-slope formula, when given a point that is the y-intercept (0,b)

slope-intercept form y = mx + b Point-slope form: y = y_1 + m(x-x_1) Substitute b for y_1 and 0 for x_1 to get the simplified point-slope form: y = b + m(x-0) and simplify y = mx + b We can use the (shorter) slope-intercept form if we know the y-intercept of the line.