Default    Projection

DrDelMath

Questions Related to Linear Functions

How do you answer these questions? Numbers refer to the question numbers above.

1. If the rule for the function is presented in the form of a first degree equation in two variables, solve for y and then replace y with f(x).

2. Plot the y-intercept (b in the form f(x) = mx + b) and one other point, then draw a line through the two points.

3.When the rule for the function is written in the form f(x) = mx + b, the y-intercept is b.

4. The zeros of any function are found by solving the equation resulting from f(x) = 0. In the case of a linear function, one must solve the equation 0 = mx + b for x.

5. Because m and b are real numbers, the solution to 0 = mx + b is a real number. Therefore the zero of a linear function is a real number and consequently is an x-intercept.

6. Whether a linear function is increasing or decreasing depends on the leading coefficient (the slope m) when the rule for f is written in the form f(x) = mx + b.

Another way of stating this is:

7. When asked to determine the rule for the linear function whose graph has a given slope and a given y-intercept, simply substitute those values into the form f(x) = mx + b for the real numbers m (slope) and b (y-intercept).

8. When asked to determine the rule for the linear function whose graph has a given slope k and contains a given point (p,q), begin by substituting the given slope k into the form f(x) = mx + b. This yields a partially determined rule containing the correct value for the slope m and an unknown value for b. Second, use the fact that if the graph contains the point (p,q), then its coordinates must satisfy the rule for the function. This yields an equation with b as the only unknown value. Solve for b and substitute it into the partially determined rule.

(This is an alternate method of answering Question 8) Use the point-slope equation for a line to write the equation of the line through the point with the given slope. Write that equation in slope-intercept form and finally replace y with f(x) to end up with proper function notation.

9. When asked to determine the rule for thelinear function whose graph contains two points, calculate the slope of the line through the two points. Use that slope and either of the two points and revert to either of the process to answer Question 8.

Examples -- Putting these Answers into Practice

Suppose the rule for the linear function is presented in the form 9x + 3y = 6
Solving for y yields y = - 3x + 2 and we replace y with f(x) to obtain the rule for the function written with standard function notation.
The y-intercept is 2 or more precisely (0,2).
The slope is -3, so the function f is decreasing.
The zero of f and its x-intercept will be found by solving 0 = f(x) = - 3x + 2.
The solution to 0 = - 3x + 2 is
Plot the points (0,2) and (,0) and draw the line through them.
The graph is shown at the right>

Find the linear function whose graph has slope 7 and y-intercept -1
Since the desired function is linear it has the form f(x) = mx + b.
In this case m = 7 and b = - 1 so the desired function is defined by the rule f(x) = 7x - 1

Find the linear function whose graph has slope 3 and contains the point (2,5).
Since the desired function is linear it must be of the form f(x) = mx + b.
In this case m = 3 and the function is partially determined by f(x) = 3x + b.
If the graph of the desired function contains the point (2,5) then 5 = f(2).
Use the partially determined rule for f (the one in boldface blue) to calculate f(2) = 3(2) + b.
This yields 5 = f(2) = 3(2) + b which is easily solved for b to obtain b = -1.
Substitute that value of b into the partially determined rule for the desired funtion (the one in boldface blue) to obtain the desired function f(x) = 3x - 1.

Find the linear function whose graph contains (2,5) and (-1,-4).
Calculate the slope of the line through these two points using the formula

to obtain m = 3.
At this point the problem may be restated as follows:
Find the linear function whose graph has slope 3 and contains the point (2,5)
This is the previous example and the problem is completed as shown in the previous example.

When asked to find the linear function whose graph contains two points, find the slope and revert to the question of finding the linear function whose graph has the calculated slope and contains either of the given points.