Problem 2043

The slope of a given line is shown below. Find the slope of a line parallel to the given line and the slope of a line perpendicular to the given line.

m = 3

 

Solution:-

 

The slope of a line parallel to the given line is m = 3

The slope of a line perpendicular to the given line is m = \frac{-1}{3}

 

Problem 2042

The slope of a given line is shown below. Find the slope of a line parallel to the given line and the slope of a line perpendicular to the given line.

m = -12

 

Solution:-

 

First find the slop of a line parallel to the given line.

Two nonvertical lines are parallel of and only of their slope are equal and they have different y-intercepts. Vertical lines are parallel if they have different x-intercepts.

Therefore, the slope of a line parallel to the given line is m = -12.

Now find the slope of a line perpendicular to the given line.

Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Put another way, two nonvertical lines are perpendicular if their slopes are negative reciprocals of each other. Any vertical line is perpendicular to any horizontal line.

To find the slop of a line perpendicular to a given line, determine the negative reciprocal of the slope of the given line.

The negative reciprocal of -12 is \frac{-1}{-12} or \frac{1}{12}.

Therefore, the slope of a line perpendicular to the given line is

m = \frac{1}{12}

 

Problem 2041

What is the range of  f(x) = 2x2 + 1.

 

Solution:-

 

The range of the function  f(x) ≥1

The graph of the function is a parabola. The positive 2 in the equation will make the graph open up. The positive one shifts the graph up vertically. The range relates to they-values with respect to the graph. Therefore, the range is y ≥1.

 

Problem 2040

What is the equation that passes through the points (4, 3) and (7, 0)?

 

Solution:-

 

Find the slope and y-intercept.

y = -x + 7

 

Problem 2039

What is the range of the following function?

f(x) = {(4,6) ,(5,7),(6,8),(7,9)}

 

Solution:-

 

Given a set of ordered pairs, the range is found by identifying the y-coordinates from the set.

So the range is {6, 7, 8, 9}.

 

Problem 2038

What is the domain of the following function?

f(x) = {(4,6) ,(5,7),(6,8),(7,9)}

 

Solution:-

 

The domain contains the x-coordinates of a set of ordered pairs.

{4,5,6,7}

 

Problem 2037

Explain why there are restrictions on the domain for the function

f(x) = \frac{1}{x}

Solution:-

 

The domain of a function is the set of values for which it is defined. For the given function x = 0 will cause the function to be undefined, since 1 divided by 0 is undefined. Therefore, the domain is the set of real numbers, except x = o or x ≠ 0.

 

Problem 2036

Explain how a graph fails the vertical line test.

 

Solution:-

 

The vertical line test is a procedure to check to see if a graph of a relation is a function. Any vertical line drawn through the graph cannot intersect the graph more than once. If any vertical line intersects more than once, the graph fails the vertical line test, and the graph is a relation. If any vertical line only intersects once, the graph passes the vertical line test, and the graph is a function.

 

Problem 2035

Explain why vertical lines have undefined slope.

 

Solution:-

 

When calculating the slope of a line, you find the change in y-coordinates divided by the change in x-coordinates. For a vertical line, the x-coordinates are constant with no change. Therefore, when you subtract their values, the difference is zero. In a fraction any number divided by zero is considered undefined. That means the slope of a vertical line is undefined.

 

Problem 2034

What is the y-intercept of the line y = -3x = 3 ?

 

Solution:-

 

The y-intercept is represented by b in y = mx + b.

-3