# College Algebra : Lines, Circles, and Piecewise Functions

## Example Questions

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### Example Question #1 : Lines, Circles, And Piecewise Functions

Which of the following will form a circle when graphed?

Explanation:

Which of the following will form a circle when graphed?

The equation of a circle follows the general formula of

Where r is the radius of the circle, and (h,k) are the coordinates of the center of the circle.

If we examine our answer choices, only

follows this form.

### Example Question #2 : Graphs

What is the center of this circle?

Explanation:

The standard form of a circle is the following:

where r is the radius and (a, b) is the center. In order to rework what we have into this form, we have to complete the square.

Since

we need to find an a so that 2a=4.

.

For the y coordinate, we need to find the same thing except instead we have the equation 2b=6.

.

And then we have the coordinates of the circle's center:

### Example Question #2 : Lines, Circles, And Piecewise Functions

Define a function

Give the -intercept of the graph of .

None of the other choices gives the correct response.

Explanation:

The -intercept of the graph of  occurs at the point at which  - that is, at the point with coordinates . By definition,  if , so we want to find this value. To do this, substitute accordingly:

Solve for  by isolating it on the left; first, subtract 15:

Divide by 7:

The -intercept of the graph of   is at .

### Example Question #3 : Lines, Circles, And Piecewise Functions

Give the -coordinate of the -intercept of the graph of the function

The graph of  has no -intercept.

The graph of  has no -intercept.

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by evaluating .

The function is piecewise-defined, so it is necessary to use the definition applicable for . However, the first definition applies for values of  less than 0, and the second, for values greater is undefined, and the graph of  has no -intercept.

### Example Question #5 : Graphs

Give the -coordinate(s) of the -intercept(s) of the graph of the function

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by solving the equation

This necessitates setting both definitions of  equal to 0 and solving for . For the first definition:

for

By the Square Root Property:

or

Since this definition holds only for , we only select .

For the second definition:

for

By the Square Root Property:

or

Since this definition holds only for , we only select .

Therefore, the graph has two -intercepts, which are at  and .

### Example Question #4 : Lines, Circles, And Piecewise Functions

Give the -coordinate(s) of the -intercept(s) of the graph of the function

The graph of  has no -intercepts.

The graph of  has no -intercepts.

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by solving the equation

This necessitates setting both definitions of  equal to 0 and solving for . For the first definition:

for

However, this definition only holds for values less than 0. No solution is yielded.

For the second definition:

for

However, this definition only holds for values greater than or equal to 0. No solution is yielded.

Therefore,  has no solution, and the graph of  has no -intercepts.

### Example Question #7 : Graphs

Give the -coordinate of the -intercept of the graph of the function

The graph of  has no -intercept.

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis. The -coordinate is 0, so the -coordinate can be found by evaluating .

The function is piecewise-defined, so it is necessary to use the definition applicable for . Since , and  for . this is the definition to use.

The -intercept of the graph is the point .

### Example Question #5 : Lines, Circles, And Piecewise Functions

True or false:

The lines of the equations

and

intersect at the point .

(Note: You are given that the lines are distinct)

True

False

True

Explanation:

If two distinct lines intersect at the point  - that is, if both pass through this point - it follows that  is a solution of the equations of both. Therefore, set  in both equations and determine whether they are true or not.

True -  is a solution of this equation.

True -  is a solution of this equation.

Therefore,  is a solution of both equations, and the lines intersect at this point.

### Example Question #6 : Lines, Circles, And Piecewise Functions

In the above diagram, the line is the graph of the equation

The circle is the graph of the equation

Graph the system of inequalities

Explanation:

The graph of an inequality that includes either the  or  symbol is the graph of the corresponding equation along with all of the points on either side of it. We are given both the line and the circle, so for each inequality, it remains to determine which side of each figure is included. In each case, this can be done by choosing any test point on either side of the figure, substituting its coordinates in the inequality, and determining whether the inequality is true or not. The easiest test point is .

This is true; select the side of this line that includes the origin.

This is true; select the side of this circle that includes the origin - the inside.

The solution sets of the individual inequalities are below:

The graph of the system is the intersection of the two sets, shown below:

### Example Question #7 : Lines, Circles, And Piecewise Functions

If these equations serve as the line of best fit for several hills, which would you LEAST like to run up?

Explanation:

The slope of a line determines its steepness.

Since slope is rise over run or

,

each of the slopes can be compared based on the ratio of rise to run.

A greater rise than run means a steeper line, or in our case, a hill. So for the "steepest" line  , one must rise 5 units and move horizontally 1 unit.

Compare this with the line  where one must rise one unit and move 8 units horizontally.

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