Algebra II : Functions and Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Graphing Linear Functions

Screen_shot_2014-12-24_at_2.55.25_pm

What is the equation of the above line?

Possible Answers:

Correct answer:

Explanation:

The equation of a line is  with m being the slope and b being the y intercept. The y-intercept is at , so . The x-intercept is , so after plugging in the equation becomes , simplifying to .

Example Question #3 : Graphing Linear Functions

Screen_shot_2014-12-24_at_2.45.09_pm

What is the equation of the line displayed above?

Possible Answers:

Correct answer:

Explanation:

The equation of a line is , with m being the slope of the line, and b being the y-intercept. The y intercept of the line is at , so .

The x-intercept is at , the equation becomes , simplification yields 

Example Question #3 : Graphing Linear Functions

Inequalities

 

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

Possible Answers:

Correct answer:

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 Linear Functions

Inequality

Which of the following inequalities is graphed above?

Possible Answers:

Correct answer:

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 Inequalities

Axes_2

Which of the following inequalities is graphed above?

Possible Answers:

Correct answer:

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 equation of the boundary:

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

 _____ 

  _____ 

 _____ 

 _____ 

1 is greater than 0 so the correct symbol is 

The inequality is 

Example Question #9 : Graphing Linear Functions

Which of the following is the function graphed below?

Graph 20150731 142249

Possible Answers:

Correct answer:

Explanation:

This function is linear (a line), so we must remember that we can represent lines algebraically using y=mx+b, where m is the slope and b is the y-intercept.

Looking at the graph, we can tell immediately that the y-intercept is -5, because the line crosses(intercepts) the y-axis at -5. 

To find the slope, we need two points,  and the following formula: 

For the sake of the example, choose (0,-5) and (2,-1). We can see that the graph clearly passes through each of these points. Any two points will do, however. Substituting each of the values into the slope formula yields m=2. 

Thus, our final answer is 

Example Question #11 : Graphing Linear Functions

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

Possible Answers:

None of these.

Correct answer:

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 #11 : Graphing Linear Functions

Where does  cross the  axis?

Possible Answers:

Correct answer:

Explanation:

To find where this equation crosses the  axis or its -intercept, change the equation into slope intercept form.

Subtract to isolate :

Divide both sides by  to completely isolate :

This form is the slope intercept form  where  is the slope of the line and  is the -intercept.

Example Question #851 : Algebra Ii

Find the -intercepts and the -intercepts of the equation.

Possible Answers:

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

To find the x-intercepts, remember that the line is crossing the x-axis, and that y=0 when the line crosses the x-axis.

So plug in y=0 into the equation above.

To find the y-intercepts, remember that the line is crossing the y-axis, and that x=0 when the line crosses the x-axis.

So plug in x=0 into the equation above.

Example Question #34 : Linear Functions

Find the slope of the line that passes through the pair of points. Express the fraction in simplest form.

 and 

Possible Answers:

Correct answer:

Explanation:

Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.

Therefore the formula for slope is as follows.

Plugging in our given points 

 and 

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