Use the equation for Budget in Exercise 23 to answer the following: a. If the slope of the equat

Use the equation for Budget in Exercise 23 to answer the...

Key Concepts

Slope

The slope of a line represents the rate of change between the two variables and measures the steepness or inclination of the line. It indicates how a change in the independent variable (often x) is reflected in a change in the dependent variable (often y). A change in the numerical value of the slope affects the line’s angle, but does not inherently change the fixed point where the line crosses the y-axis unless other parameters of the equation are modified.

Vertical Intercept

The vertical intercept, or y-intercept, is the point where the line crosses the y-axis. It represents the value of the dependent variable when the independent variable is zero. In the context of a linear equation in slope-intercept form, this intercept remains constant unless a specific change is made to the constant term of the equation. Therefore, altering only the slope does not directly impact the vertical intercept.

Horizontal Intercept

The horizontal intercept, or x-intercept, is the point at which the line crosses the x-axis and it is found by setting the dependent variable to zero in the equation. Changes in the slope can affect the horizontal intercept because they alter the rate at which the graph descends or ascends. A modification in the slope while keeping the intercept fixed typically results in a new x-intercept position, reflecting the reorientation of the line's path across the graph.

Transcript

00:01 In the given question we are told that we have to answer two questions regarding a certain equation that is given to us in the question.

00:13 So the equation goes like this.

00:17 It says that b is equal to 0 .55m plus 29 .95 is an equation that represents the cost in dollars for renting a 10 -foot truck from budget and driving it m miles.

00:45 So we can write over here that b is the cost in dollars of renting a 10 -foot truck and is the number of miles driven.

01:28 So this equation gives us the cost of renting a 10 -foot truck and it calculates according to the equation 0 .55m plus 29 .95.

01:44 And in the question what we are asked regarding this equation is that if the slope of this equation is changed to 0 .75 how would that affect the vertical intercept of the graph.

02:00 So what is the vertical intercept of the graph is what we should find first.

02:06 So we can write to find the vertical intercept.

02:11 The vertical intercept is the point where the graph intersects the y -axis which means it is a point on the y -axis and the points any points on y -axis would have its x -coordinate as 0.

02:30 So to find the vertical intercept what we should do is we should simply substitute the value of the x -coordinate value of x -coordinate x -coordinate x -coordinate as 0.

02:53 So what would happen then is when we take let's say the x -coordinate over here is represented by is represented by m right m is the x -coordinate equivalent of x -coordinate in this equation.

03:16 So when m is equal to 0 m is equal to 0 we can write b is equal to 0 .55 0 .55 times 0 plus 29 .95.

03:39 So we get the corresponding value of b when m equal to 0 it is 29 .95.

03:46 Right.

03:47 So now what we could do is we can write that the y -intercept or the vertical intercept vertical intercept is having an x -coordinate of 0 and a y -coordinate of 29 .95.

04:11 And now we are told that if the slope of this graph is changed right if the slope is changed to 0 .75 what would be the change on the vertical intercept.

04:28 So one thing that we should know about this equation is that this equation is in the form that we call the slope intercept form.

04:40 So what is the slope intercept form.

04:42 So in slope intercept form the equation will be of the form y equals m times x plus c.

04:51 And since we have already used m in this equation let's use another letter over here.

04:56 Let's say we are going to use k.

05:00 Right.

05:00 Y is equal to k x plus c is the format of a slope intercept form and over here the coefficient of x -coordinate or x the variable x in this equation would give us the slope of the line and the constant in this equation will give us the y -intercept.

05:22 So how does it give us the y -intercept.

05:25 It is the y -intercept or the vertical intercept as we can take if you know the value of c you can take the vertical intercept as 0 comma c.

05:37 So let's check that with the equation that we have.

05:41 Right.

05:41 So the slope of the equation that we have will be 0 .55 and the vertical intercept would be at 029 .95.

05:56 So that's exactly what we have got over here.

05:59 So now if we have to change the slope of the equation that we have in this question what we can do is we can write then b is equal to 0 .75 m plus 29 .95.

06:22 Right.

06:22 If the slope is changed this is what would happen.

06:26 So now you can take to find the vertical intercept.

06:34 The vertical intercept what we could do is we can substitute m equals 0 and what you would have then is b is equal to 0 .75 times 0 plus 29 .95.

06:56 So that gives you the value of b as 29 .95 which means the vertical intercept is still the vertical intercept intercept is still is still 029 .95.

07:19 Right.

07:19 So the answer to this question is that there is no change in the vertical intercept.

07:26 There is no change in the vertical intercept when the graph of this equation is changed.

07:35 When the slope of this equation is changed.

07:40 Slope of this graph is changed.

07:42 So this is what we had to find in this in the first part of this question...

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