Linear Functions

Linear Functions

Domain, Range, Slope Intercepts

Linear Functions -> y=mx+b

• m is for Slope, or Rate of Change

• b is for the Y-intercept

What does Slope (m) mean again ?

The number value with slope means how quickly something changes. The larger the number, the faster the rate of change, the smaller, the slower the rate of change.

Slope also represents the steepness of a graph.

Slope is also  $\frac{Rise}{Run}$   

Another way to look at slope is Change of the y values over the change of x values.
In most real world situations the 'Run' is represented with time.

With a fraction, you can divide the two numbers, since a fraction is a representation of division, to make it easier to compare slopes.

Multiple Choice

Which of the numbers represent the greatest rate of change (slope)?

• $\frac{6}{5}$

• $\frac{3}{1}$

• $\frac{9}{4}$

• $\frac{12}{6}$

Sometimes with slope, you can set up a proportion to solve.

• Cross multiply and divide.

Multiple Choice

The total distance in centimeters a toy robot moves varies directly with the time in seconds. The toy robot moves a total distance of 264 centimeters in 11 seconds.

What is the time in seconds the toy robot moves when the total distance is 408 centimeters?

• 24 s

• 17 s

• 13 s

• 37 s

What is the y-intercept (b) ?

Y-intercept is literally where the graph crosses the y-axis.

In real world scenarios, it represents a starting point or value.

For example, if you are going to start saving money, the y-intercept would represent how much money you start with in your account.

Multiple Choice

What would the y-intercept be on the graph?

• 5

• 3

• 6

• 1

You can determine the equation from a graph using the point-slope form

•  $y-y_1=m\left(x-x_1\right)$  

• Count Slope - with Rise over Run

• Find a point on the graph and plug in the x and the y

On the following problem.....

• Count Rise and Run starting from the y-intercept, which is as (0, -3) to point (7, 2)

• Using the point (7, 2) plug the x and y into the point-slope form.

Multiple Choice

Which equation is best represented by the graph?

• $y+2=\frac{7}{5}\left(x+7\right)$

• $y-2=\frac{7}{5}\left(x-7\right)$

• $y+2=\frac{5}{7}\left(x+7\right)$

• $y-2=\frac{5}{7}\left(x-7\right)$

Plugging into point slope form

plug in, then solve for y to get it into slope-intercept form

Use point-slope on the following problem

•  $y-y_1=m\left(x-x_1\right)$  

• The problem crosses the x-axis at 36, so the (x, y) you use is (36, 0)

• perpendicular is the opposite reciprocal.  

• the slope is -4/9, so the opposite reciprocal is 9/4

Multiple Choice

What is the equation of a line in slope-intercept form that crosses the x-axis at 36 and is perpendicular to the line

 $y=-\frac{4}{9}x+5$  

•  $y=\frac{4}{9}x+16$  

•  $y=\frac{4}{9}x-16$  

•  $y=\frac{9}{4}x+81$  

•  $y=\frac{9}{4}x-81$  

Sometimes you need to find the slope also given two points. Use slope formula

 $\frac{y_2-y_1}{x_2-x_1}$  

Find slope first on the next problem then plug into point slope.

Multiple Choice

What is the equation in slope-intercept form of the line that passes through the points (-4, 2) and (12, 6)?

• $y=0.25x+3$

• $y=0.25x-4.5$

• $y=4x+18$

• $y=4x-42$

You can set up a system of equations to solve for multiple variables.

• Solve for one variable

• Then plug in and solve for the other

Set up a system to solve the following scenario

• A college student completed some courses worth 3 credits and some courses worth 4 credits. The student earned a total of 59 credits after completing 18 courses. How many courses worth 3 credits did the student complete?

• First, we need to write two equations to solve.

• He completed 18 courses, so the first equation is x + y = 18

• The x represents the 3 hour courses and y for the 4 hour courses.

Set up the second equation to solve

• First is x + y = 18

• He completed 59 credits, so..

• 3x + 4y = 18 is the second equation

• how many 3 hour courses times 3 to get those credits how many 4 hour courses times 4 for those credits.

Now Solve

Find one, then plug in to get the other.

Multiple Choice

So if x represents the number of 3 hour courses and y represents the number of 4 hour courses, How many courses worth 3 hour credits did he complete?

• 13

• 5

• 20

• 39

Now complete the following questions that use what we just covered.

Multiple Choice

Which value of x makes the equations 1.25(4x-10)=7.5 true? (hint: solve for x)

• 3.5

• -1

• -0.5

• 4

Multiple Choice

Which answer choice best describes the domain (x) and range (y) of the function for this situation?

• Domain: ARN < 100

  Range: ARN < 50

• Domain: { -2 }

  Range: { 100 }

• Domain: ARN < 50

  Range: ARN < 100

• Domain: { 100 }

  Range: { -2 }

Multiple Choice

A grill at a barbecue restaurant will be used to cook sausage links that are 2 lbs each and briskets that are 6 lbs each. No more than 120 lbs of sausage and briskets will be cooked.

Which represents the combinations of x, the sausage, and y, the briskets?

• $6x+2y<120$

• $2x+6y\le120$

• $6x+2y>120$

• $2x+6y\ge120$

Multiple Choice

Which system represents the graph?

• 
  

  $y=-x-4$
  
  $y=2x-2$
  

• $y=-x+4$
  $y=2x-4$
  

• $y=x-4$
  $y=-2x-2$
  

• $y=x+4$
  $y=-2x-4$
  

Multiple Choice

Which equation represents the table? (hint: find slope, then the y-intercept. When x is 0 what is y?)

• $f\left(n\right)=-3n+24$

• $f\left(n\right)=-\frac{1}{3}n+16$

• $f\left(n\right)=-3n+64$

• $f\left(n\right)=-\frac{1}{3}n+8$

Multiple Choice

What is the value of x to this system of equations?

3x - 5y = 22
y = -5x + 32

• -6.5

• 0.5

• 6.5

• -0.5

Multiple Choice

What is the solution to -(6m + 8) = 4(17 - m) ? (hint: solve for m)

• 76

• 38

• 2

• 6

Multiple Choice

What is the equation that is parallel to y = 1.2x + 3.8 and passes thru the point ( 5, 0 ) ? (hint: plug into point slope form)

• 
  

   $y=1.2x-6$  
  

•  $y=-1.2x+6$  

•  $y=1.2x+5$  

•  $y=-1.2x-5$  

Multiple Choice

A customer at a store paid $64 for 3 large candles and 4 small candles. At the same store, a second customer paid $4 more than the first customer for 1 large candle and 8 small candles. The price of each large candle and each small candle is the same. Which system can used to find the price of each large, x, candle and each small, y, candle?

• 
  

  $4y=3x+64$
  

  
  $8y=x+68$
  

• $4y=3x+64$
  $8y=x+60$
  

• $3x+4y=64$
  $x+8y=68$
  

• $3x+4y=64$
  $x+8y=60$