SAT II Math I : Graphing Linear Functions

Example Questions

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Example Question #1 : Graphing Functions

Refer to the above red line. A line is drawn perpendicular to that line, and with the same -intercept.  Give the equation of that line in slope-intercept form.

Explanation:

First, we need to find the slope of the above line.

The slope of a line. given two points  can be calculated using the slope formula

Set :

The slope of a line perpendicular to it has as its slope the opposite of the reciprocal of 2, which would be . Since we want this line to have the same -intercept as the first line, which is the point , we can substitute  and  in the slope-intercept form:

Example Question #1 : Graphing Linear Functions

Refer to the above diagram. If the red line passes through the point , what is the value of ?

Explanation:

One way to answer this is to first find the equation of the line.

The slope of a line. given two points  can be calculated using the slope formula

Set :

The line has slope 3 and -intercept , so we can substitute  in the slope-intercept form:

Now substitute 4 for  and  for  and solve for :

Example Question #119 : Functions And Graphs

Which equation best matches the graph of the line shown above?

Explanation:

To find an equation of a line, we will always need to know the slope of that line -- and to find the slope, we need at least two points. It looks like we have (0, -3) and (12,0), which we'll call point 1 and point 2, respectively.

Now we need to plug in a point on the line into an equation for a line. We can use either slope-intercept form or point-slope form, but since the answer choices are in point-slope form, let's use that.

Unfortunately, that's not one of the answer choices. That's because we didn't pick the same point to substitute into our equation as the answer choices did. But we can see if any of the answer choices are equivalent to what we found. Our equation is equal to:

which is the slope-intercept form of the line. We have to put all the other answer choices into slope-intercept to see if they match. The only one that works is this one:

Example Question #1 : Graphing Linear Functions

Determine where the graphs of the following equations will intersect.

Explanation:

We can solve the system of equations using the substitution method.

Solve for  in the second equation.

Substitute this value of  into the first equation.

Now we can solve for .

Solve for  using the first equation with this new value of .

The solution is the ordered pair .

Example Question #2 : Graphing Linear Functions

Refer to the line in the above diagram. It we were to continue to draw it so that it intersects the -axis, where would its -intercept be?

Explanation:

First, we need to find the slope of the line.

In order to move from the lower left point to the upper right point, it is necessary to move up five units and right three units. This is a rise of 5 and a run of 3. makes the slope of the line shown .

We can use this to find the -intercept  using the slope formula as follows:

The lower left point has coordinates . Therefore, we can set up and solve for  in this slope formula, setting :

Example Question #3 : Graphing Linear Functions

Line  includes the points  and . Line  includes the points  and . Which of the following statements is true of these lines?

The lines are distinct but neither parallel nor perpendicular.

The lines are identical.

The lines are parallel.

Insufficient information is given to answer this question.

The lines are perpendicular.

The lines are parallel.

Explanation:

We calculate the slopes of the lines using the slope formula.

The slope of line  is

The slope of line  is

The lines have the same slope, making them either parallel or identical.

Since the slope of each line is 0, both lines are horizontal, and the equation of each takes the form , where  is the -coordinate of each point on the line. Therefore, line  and line  have equations  and .This makes them parallel lines.

Example Question #4 : Graphing Linear Functions

Refer to the above diagram. which of the following compound inequality statements has this set of points as its graph?

Explanation:

A horizontal line has equation  for some value of ; since the line goes through a point with -coordinate 3, the line is . Also, since the line is solid and the region above this line is shaded in, the corresponding inequality is .

A vertical line has equation  for some value of ; since the line goes through a point with -coordinate 4, the line is . Also, since the line is solid and the region right of this line is shaded in, the corresponding inequality is .

Since only the region belonging to both sets is shaded - that is, their intersection is shaded - the statements are connected with "and". The correct choice is .

Example Question #1 : Graphing Inequalities

Which of the following inequalities is graphed above?

Explanation:

First, we determine the equation of the boundary line. This line includes points  and  , so the slope can be calculated as follows:

Since we also know the -intercept is , we can substitute  in the slope-intercept form to obtain the equation of the boundary line:

The boundary is included, as is indicated by the line being solid, so the equality symbol is replaced by either  or . To find out which one, we can test a point in the solution set - for ease, we will choose :

_____

_____

_____

0 is less than 3 so the correct symbol is

The inequality is .

Example Question #1 : Graphing Linear Functions

Select the equation of the line perpendicular to the graph of .

None of these.

Explanation:

Lines are perpendicular when their slopes are the negative recicprocals of each other such as . To find the slope of our equation we must change it to slope y-intercept form.

Subtract the x variable from both sides:

Divide by 4 to isolate y:

The negative reciprocal of the above slope:  . The only equation with this slope is

Example Question #7 : Graphing Linear Functions

An individual's maximum heart rate can be found by subtracting his or her age from . Which graph correctly expresses this relationship between years of age and maximum heart rate?