I have a few questions that I don't understand hope someone can help.
1)write an equation of th line, in point slope form, that passes through the two given points. (-17,8),(3,-2)
I figured (y-8)=-2(x+17) not sure
2) the graph of g(x) is f(x) vertically compressed by a factor of 1/7 and reflected in the x-axis what is the function rule for g(x)given f(x)=2x.
not sure about this one
3)determine whether y varies directly with x if so find the constant of variation k and write the equation.
x y
7 11
8 13
9 15
10 17
I know you start with y=kx but then I'm lost.
not looking for just answers but if someone can explain I'd appreciate it
on number 3 y=kx
so would i divide y by x and if so I get 1.57, 1.62, 1.66, and 1.7..Is this right?
#1. If not sure, plug in the points and see whether they satisfy your equation. you will see that they don't even come close
The (y-8) and (x+17) bits are ok, but how did you come up with a slope of -2? (-2-8)/(3+17) = -1/2
Looks like you used ∆x/∆y for the slope, rather than ∆y/∆x!
#2: you want g(x) which produces a height 1/7 of f(x). So,
g(x) = 1/7 f(x)
Now, you want to reflect g(x) around the x-axis, so
h(x) = -g(x)
h(x) = -1/7 f(x) = -1/7 (2x) = -2/7 x
check: f(1) = 2
h(1) = -2/7 shrunk, reflected
that is correct. Since the ratio changes, it is not a constant k.
y does not vary directly with x
In fact, y increases by 2 for every increase in x by 1. So,
y = 2x-3
That pesky -3 fouls up the direct variation.
1) To find the equation of a line in point-slope form, you need the coordinates of two points on the line. Let's use the points (-17, 8) and (3, -2).
The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.
First, let's find the slope (m). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1).
Using our given points (-17, 8) and (3, -2), the slope is: m = (-2 - 8) / (3 - (-17)) = -10 / 20 = -1/2.
Now, let's choose one of the points (let's use (-17, 8)) and substitute its coordinates into the point-slope form equation:
y - y1 = m(x - x1)
y - 8 = (-1/2)(x - (-17))
(y - 8) = (-1/2)(x + 17)
So, the equation of the line in point-slope form that passes through the points (-17, 8) and (3, -2) is (y - 8) = (-1/2)(x + 17).
2) To vertically compress a function by a factor of 1/7, you multiply the function's output (y-values) by 1/7. To reflect a function in the x-axis, you multiply the whole function by -1.
Given f(x) = 2x, let's first vertically compress it by a factor of 1/7: g(x) = (1/7)(2x) = (2/7)x.
Now, let's reflect it in the x-axis by multiplying the function by -1: g(x) = -(2/7)x.
Therefore, the function rule for g(x) is g(x) = -(2/7)x.
3) To determine whether y varies directly with x, you need to check if the ratio between y and x is constant for all the given data points.
Let's calculate the ratios for the given data points:
- For the first data point (7, 11), the ratio is 11/7 = 1.57.
- For the second data point (8, 13), the ratio is 13/8 = 1.63.
- For the third data point (9, 15), the ratio is 15/9 = 1.67.
- For the fourth data point (10, 17), the ratio is 17/10 = 1.70.
Since the ratios are approximately constant (1.57, 1.63, 1.67, 1.70), we can conclude that y varies directly with x.
Now, to find the constant of variation (k), we can choose any of the data points. Let's use (7, 11) as an example.
Using the formula y = kx, we substitute the given coordinates (7, 11):
11 = k * 7
Solving for k, we divide both sides by 7:
k = 11 / 7 = 1.57
The constant of variation (k) is approximately 1.57.
Therefore, the equation for this direct variation is y = 1.57x.