Linear Relations (extended version)

How do we graph using the intercept method? 2x + y = 8

1. Find the x-intercept (sub y = 0) 2x + 0 = 8 2x ÷ 2 = 8 ÷ 2 x = 4 ∴ x-intercept is 4 and coordinate pair is (4,0) 2. Find the y-intercept (sub x = 0) 2(0) + y = 8 y = 8 ∴ y-intercept is 8 and coordinate pair is (0,8) 3. Graph the two coordinate pairs.

What are the following equations in words: 1. C = 10n +50 (if C is cost in dollars and n is number of t-shirts sold) 2. T = 0.5t + 13 (if T is Temperature in degrees and t is time in minutes a solution is heated)

1. The total cost equals $10 per number of t-shirts plus an initial cost of $50. 2. The temperature equals 0.5° per min a solution is boiled plus an initial temperature of 13°.

What is the dependent variable?

The y variable; it depends on the x variable.

How do you find the initial value by looking at a table of values?

What is y when x is 0.

Does the line 2x+3y = 6 contain the point (18,10)?

Yes

How do you find rate of change by looking at a table of values?

∆y÷∆x (NOTE: ∆ means change in)

What quadrant would the following points be in? (2, 3) (-2, 3) (2, -3) (-2, -3) (0,4)

(2, 3) - quadrant I (-2, 3) - quadrant II (2, -3) - quadrant IV (-2, -3) - quadrant III (0,4) - n/a

If asked to find the coordinates, how must I write it?

(x,y)

What does the coefficient in the equation tell you? y = 3x+1

1. The steepness of the graph 2. The amount y changes everytime x increases by 1.

Complete a table of values and identify the equation for the following: Crothall Car Rental charges customers $20 per day, plus $0.10 for each km driven.

C = 20 + 0.10n

A negative coefficient means the slope is...

Decreasing

When would you use a dotted line to connect the dots on a graph?

If you have some of the values between points but not all.

What is the y-intercept?

On a graph: where the line intersects with the y-axis. On a table of values: what is y when x is 0. On an equation: substitute 0 for x to find the y-intercept. In a coordinate pair: (0,?)

1600 m (extrapolation)

Predict how far she can jog in 13 min. Is this interpretation or extrapolation?

15 min (extrapolation)

Predict how long it will take her to jog 1800m. Is this interpretation or extrapolation?

Determine a coordinate on the line 2x+3y = 6

Sub x = any # you choose

y = (rate of change)(x) + initial value D = 15n+0 D = 15n

Substitute the following into the standard equation to find the equation for the Table of Values: 1. label for x 2. label for y 3. rate of change (15) 4. initial value (0)

How do you find the x-intercept? 2x + y = 8

Substitute y=0 into the equation 2x + 0 = 8 2x ÷ 2 = 8 ÷ 2 x = 4 ∴ x-intercept is 4 and coordinate pair is (4,0)

Switch to definition first

Switch to definition first

Switch to starting with definition first

Switch to starting with definition first

Switch to starting with term first.

Switch to starting with term first.

What is the constant in a linear equation y = 3x+6

The constant is the same as the y-intercept (the initial value; what is y when x is 0?)

What does the +50 (an initial cost of $50) and the +13 (an initial temperature of 13°) represent?

The initial value of y when x is 0. Examples 1. The initial cost of printing the t-shirts (y) even if you buy 0 t-shirts (x) - because of the cost of silkscreening. 2. The temperature (y) when you have boiled a solution for 0 minutes (x).

Define interpolation

predicting values that lie between data points on a graph (inter = inside; between)

Define extrapolation

predicting values that lie outside data points on a graph (extra = beyond; outside)

Use table of values method to graph y = -3x + 6

substitute a couple of values of x into the equation to find the y coordinate. NOTE: use some negative values and some positive values if no context is given.

Plot and label the following points on the grid the origin (-3, 1)

the origin (0,0)

The equation x = 2 means...

x is always 2 (no matter what y is).

Where are points that have a y-coordinate of 0 located?

x-axis

What is the standard form of a linear equation?

y = (rate of change)(x) + initial value

The equation y = -3 means...

y is always -3 (no matter what x is).

Where are points that have an x-coordinate of 0 located?

y-axis

What is the independent variable?

The x variable; the y changes because of it (often time and number).

Rate of change = ∆y÷∆x Rate of change = 15÷1 Rate of change = 15 Initial value: 0

Identify the rate of change and the initial value for the following Table of Values.

1. Rate of change = ∆y÷∆x Rate of change = 10÷1 Rate of change = 10 2. Initial value: 50 3. Equation: y = (rate of change)(x) + initial value C = 10n + 50

Identify the rate of change the initial value and the equation for the following Table of Values.

1. Rate of change = ∆y÷∆x Rate of change = 4÷2 Rate of change = 2 2. Initial value: 2 3. Equation: y = (rate of change)(x) + initial value p = 2n + 2

Identify the rate of change the initial value and the equation for the following Table of Values.

When can you connect the dots on a graph?

If your data is continuous (you can have all values in between the dots) Example: You can drive 10.2 km and pay $13.72 for gas.

When do you use dots when graphing a relation?

If your data is discrete (you can not have fractions or decimals) Example: You can't sell fractions of a t-shirt

A positive coefficient means the slope is...

Increasing

850m (Interpolation)

Julia is jogging along the Galloping Goose Trail. Her trainer has graphed her progress. Estimate how far Julia can jog in 7 min. Is this interpretation or extrapolation?

What is a horizontal line?

On a graph: A line straight across In an equation: there is no x in equation In a table of values: y always has the same value no matter what x is.

What is a vertical line?

On a graph: A line straight up and down In an equation: there is no y in equation In a table of values: x always has the same value no matter what y is.

What is an oblique line?

On a graph: A slanted line In an equation: equation contains both an x and y variable)

What is the x-intercept?

On a graph: where the line intersects with the x-axis. On a table of values: what is x when y is 0. On an equation: substitute 0 for y to find the x-intercept. In a coordinate pair: (?,0)

What are the following equations in words: 1. C = 10n (if C is cost in dollars and n is number of t-shirts sold)

The total cost equals $10 per number of t-shirts

What does the 10n ($10 per t-shirt) represent?

The rate of change (how much y changes with a 1 unit change of x) Example: How much the cost (y) increases with the increase of 1 t-shirt (x).