In the linear function h, h(0) = 41 and h(1) = 40

Question

Write an equation defining h.

Answer 1

Given: The linear function is passing through two points (x₁, h₁) = (0, 41) and (x₂, h₂) = (1, 40).

To find the equation of a linear function from two points, use proportionality:

h - h₁ x - x₁

---------- = ---------

h₂ - h₁ x₂ - x₁

h - 41 x - 0

---------- = -------- cross-multiply => h - 41 = -x => h = -x + 41

40 - 41 1 - 0

Answer: h = -x + 41

Answer 2

The y-intercept occurs when the x coordinate is 0. Since we know that h(0) = 41, we know that the y-intercept is 41 and can therefore solve this question using slope-intercept form.

The formula for slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

We already have the y-intercept (41), so now we need to figure out the slope. The slope formula is the change in y divided by the change in x. Thus, we need to find the value of (y₂ - y₁)/(x₂ - x₁). In this case, let's say that (0,41) is coordinate 1 (so x₁ = 0 and y₁ = 41) and (1,40) is coordinate 2 (so x₂ = 1 and y₂ = 40). So, we get (40-41)/(1-0) = (-1)/1 = -1. We solved for the slope and found that it equals -1.

Now, let's put it all together. We will take the values we found and plug them into the general slope-intercept form y = mx + b. y = (-1)x + 41 or y = -x + 41.

Hope this helps!

Answer 3

Consider the two values of h(0)=41 and h(1)=40.

By definition, the slope-intercept form is y=mx+b where m is the slope given by m=(y₂-y₁)/(x₂-x₁).

Then we have x₁=0, y₁=41, x₂=1, y₂=40.

This gives m=(40-41)/(1-0)=-1/1, so the slope is -1.

Observe that y=-x+b where b=41 given that h(0)=41, since x₁=0.

Thus y=-x+41.

Therefore, in the linear function h where h(0)=41 and h(1)=40, an equation defining h is h=-x+41.

Answer 4

Slope-intercept form: y = mx + b

For simplicity, we can rename h(x) as y.

h(0) = 41, which means that when x = 0, y = 41

h(1) = 40, which means that when x = 1, y = 40

To calculate the slope (m), divide the change in y by the change in x.

m = (41 - 40)/(0 - 1) = (1)/(-1) = -1

We now know that y = -x + b

The value of y when x = 0 is known as the y-intercept (b)

We are given that y = 41 when x = 0, so b = 41.

We now know that y = -x + 41

We can rewrite this as h(x) = -x + 41, which is our final answer!

Answer 5

y = mx + b is a linear function with slope m and y intercept b.

It can be written as h(x) = y = mx + b or just h(x) = mx + b

h(0) = 41 = m*0 + b

41 = b

h(x) = mx + 41

h(1) = 40 = m*1 + 41

-1 = m

h(x) = -x + 41 is the equation.

Answer 6