Linear Functions - Pre-Calculus

First, we need to compute , the slope. We can do this with the slope formula

, sometimes called "rise over run"

So we now have

Now in order to solve for we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use .

We then have:

Which becomes .

Hence we take our found value for and plug it back into to get

.

For the linear function in point-slope form

The slope is equal to

For this problem

we get

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point in for and giving .

This will simplify to , or after subtracting the fraction.

Hence our answer is after plugging our values for and in.

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them. .

.

Now use the point slope formula:

distribute the right side

add 6 to both sides

which is the final answer.

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

To solve for the equation of this problem, we will need 2 points. Let the point be defined as , where is time in hours and represents the dollars Billy has earned.

Billy starts with three dollars in the first hour. Write the point.

At the end of eight hours, Billy has fifty dollars. Write the second point.

Write the slope-intercept formula.

Write slope formula.

Plug a point and the given slope to the slope-intercept formula to solve for the y-intercept.

Substitute this and the slope back into the slope-intercept equation.

Since Billy's earning is dependent on the time he works, rewrite the equation so that time is the independent variable, and amount earned is the dependent.

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

First find the reciprocal of the slope of the given function.

The perpendicular function is:

Now we must find the constant, , by using the given point that the perpendicular crosses.

solve for :

For the linear function in point-slope form

The slope is equal to

For this problem

we get

Which of the following could be the function modeled by this graph?

We can begin here by trying to identify a couple points on the graph

We can see that it crosses the y-axis at

Therefore, not only do we have a point, we have the y-intercept. This tells us that the equation of the line needs to have a in it somewhere. Eliminate any option that do not have this feature.

Next, find the slope by counting up and over from the y-intercept to the next clear point.

It seems like the line goes up 5 and right 1 to the point

This means we have a slope of 5, which means our equation must look like this:

By definition, the y-intercept is the point on the line that crosses the y-axis. This can be found by substituting into the equation. When we do this with our equation,

.

Alternatively, you can remember form, a general form for a line in which is the slope and is the y-intercept.

Recall slope-intercept form, or . In this form, is the slope and is the y-intercept. Given our equation above, the slope must be the coefficient of the x, which is .

The x-intercept of an equation is the point at which the line crosses the x-axis. Thus, we can find the x-intercept by plugging in . When we do this with our equation:

Thus, our x-intercept is the point .

First, we need to compute , the slope. We can do this with the slope formula

, sometimes called "rise over run"

So we now have

Now in order to solve for we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use .

We then have:

Which becomes .

Hence we take our found value for and plug it back into to get

.

For the linear function in point-slope form

The slope is equal to

For this problem

we get

Since our slope is , we can plug it into right away giving .

To solve for , we plug our given point in for and giving .

This will simplify to , or after subtracting the fraction.

Hence our answer is after plugging our values for and in.

Linear functions follow the form , where m is the slope and b is the y intercept. We can determine the equation of a linear function when we have the slope and a y intercept which is a starting point for drawing our line. If we need the equation of a line that passes through two points we use the point slope equation:

where the variables with numbers next to them correspond to places where you input the x and y from a single (same) coordinate pair.

Since we have two points, we can compute the slope of a line that passes between them. .

.

Now use the point slope formula:

distribute the right side

add 6 to both sides

which is the final answer.

Take a look at the coordinate pairs. Was it necessary to to use the point slope formula? The answer is no, since the coordinate pair (0,5) is already a y intercept. So all that was necessary was to compute the slope.

Bob starts out with 4 candies. Write the equation.

Every week, he earns 14 candies. Every week has a total of 7 days. Divide the amount of candies by the number of days to determine the number of candies Bob earn in a day.

Bob then earns 2 candies per day. Let represent the number of candies per day. Finish the incomplete equation.

To solve for the equation of this problem, we will need 2 points. Let the point be defined as , where is time in hours and represents the dollars Billy has earned.

Billy starts with three dollars in the first hour. Write the point.

At the end of eight hours, Billy has fifty dollars. Write the second point.

Write the slope-intercept formula.

Write slope formula.

Plug a point and the given slope to the slope-intercept formula to solve for the y-intercept.

Substitute this and the slope back into the slope-intercept equation.

Since Billy's earning is dependent on the time he works, rewrite the equation so that time is the independent variable, and amount earned is the dependent.