I've tried so hard on this and I'm completely hopeless, can somebody help me out? Please it would mean a lot. Task 1 find the rate of change for each table, equation, or graph. How did you find the rate of change?
Task 1
A.
X 2 4 6
Y 10 20 30
Rate of change:
How did you find the rate of change:
B.
5x +3y= -2
Rate of change:
How did you find the rate of change:
Write an equation in slope-intercept, point-slope or standard form for the line with the given information. Explain why you chose the form you used.
Task 2
A.
Passes through (-1,4), (-5,2)
Equation:
Explain why you chose the form you used:
B.
Slope 2 , y-intercept -4
Equation:
Explain why you chose the form you used:
C.
Has an x-intercept of 6 and a y-intercept of 3
Equation:
Explain why you chose the form you used:
D.
Passes through (1,2) with slope
Equation:
Explain why you chose the form you used:
Please help :(
the rate of change is the slope of the line. That is, if x changes by 1, how much does y change?
In the slope-intercept form, the rate of change (slope) is m, in
y = mx+b
The point-slope form of a line is
y-k = m(x-h)
where the line passes through (h,k) with slope m. It's just another way of writing
(y-k)/(x-h) = m
That is, the slope (m) is the ratio of the change in y over the change in x.
I'm sure all of this is explained in your text, with examples.
So, with all this info, you should have a handle on solving the problems you posted.
If you are given a slope and an intercept, choose the slope-intercept form.
If you are given a point and a slope, use the point-slope form.
If you are given two points, figure the slope, then use one of the points and the point-slope form.
I'm here to help! Let's tackle Task 1 first.
A. To find the rate of change for a table, equation, or graph, you need to determine how much the dependent variable (Y) changes for every one unit increase in the independent variable (X). In this case, we have the following table:
X: 2, 4, 6
Y: 10, 20, 30
To find the rate of change, we calculate the difference in the Y values and divide it by the difference in the X values. In this case, we can see that every time X increases by 2, Y increases by 10. Therefore, the rate of change is 10/2 = 5.
B. The given equation is 5x + 3y = -2. To find the rate of change from an equation, we can rewrite it in slope-intercept form (y = mx + b), where "m" represents the rate of change (slope). To do this, isolate y in terms of x:
5x + 3y = -2
3y = -5x - 2
y = (-5/3)x - (2/3)
From the equation, we can see that the rate of change is -5/3.
Now, let's move on to Task 2.
A. To write an equation when given two points, we can use the point-slope form (y - y1 = m(x - x1)), where (x1, y1) is a known point on the line and m is the slope. Using the points (-1,4) and (-5,2), we can calculate the slope:
m = (y2 - y1) / (x2 - x1)
= (2 - 4) / (-5 - (-1))
= -2 / (-4)
= 1/2
Now we have the slope (m = 1/2). We can choose to use either point-slope form or slope-intercept form. Let's use point-slope form:
y - y1 = m(x - x1)
y - 4 = (1/2)(x - (-1))
y - 4 = (1/2)(x + 1)
y - 4 = (1/2)x + 1/2
This is an equation in point-slope form.
B. When given the slope (m = 2) and the y-intercept (y-intercept = -4), we can directly write the equation in slope-intercept form (y = mx + b). Plugging in the given values, we have:
y = 2x - 4
This is an equation in slope-intercept form.
C. When given the x-intercept (6) and the y-intercept (3), we can write the equation in standard form (Ax + By = C), where A, B, and C are coefficients. The x-intercept implies that the y-coordinate is 0, and the y-intercept implies that the x-coordinate is 0.
Using the intercepts, we have:
x-intercept: (6, 0)
y-intercept: (0, 3)
Now we can plug these points into the standard form equation and solve for A, B, and C:
6A + 0B = C
0A + 3B = C
From these equations, we can see that A = C and B = C/3. Let's assign C = 3 (arbitrarily chosen). Then, A = 3 and B = 1. The equation becomes:
3x + y = 3
This is an equation in standard form.
D. When given a point (1,2) and the slope, the best choice of form depends on the given information. If the slope is known, we can use point-slope form or slope-intercept form. Let's choose slope-intercept form this time.
Given the point (1,2) and the slope (unknown), we can plug in the point coordinates into the slope-intercept form equation (y = mx + b) and solve for b:
2 = m(1) + b
2 = m + b
b = 2 - m
Now we have the slope-intercept form equation:
y = mx + (2 - m)
This is an equation in slope-intercept form.
I hope this explanation helps you understand how to find the rate of change and write equations in different forms. If you have any further questions, feel free to ask!