Slopes (Again) - At A Glance

The line below has a slope of 5.

Fill in the indicated number.

The missing number is Δ y and we're told Δ x = 2.

This means the missing number is

Δ y = slope × Δ x = (5)(2) = 10.

The line below has a slope of 4.

Fill in the indicated number.

The line has slope 4 and Δ x = 1.

So

Δ y = slope × Δ x = 4 × 1 = 4.

The number we want is the sum of Δ y and the value of y at the first point. therefore the missing number is

4 + 1 = 5.

When doing problems like this, it can be helpful to think about the values as y_old and y_new.

y_old is usually the value of y we're given in the problem. If we add y_old and the change in y, we get y_new.

y_new = y_old + Δ y.

The line below has a slope of -0.7. Find the indicated number.

We have y_old = 3 and we want to find y_new. We know

y_new = y_old + Δ y,

so we just need to find Δ y, which we know how to do:

Δ y = slope × Δ x = (-0.7)(4.5) = -3.15.

Now we know

y_new = 3 + (-3.15) = -0.15.

The function f is a line with slope . If f (a) = 3, what is ?

The function f is a line with slope that passes through the point (a, 3). The old value of f is 3 and we want to know the new value of f if we change x by :

We don't know what a is, but that's ok. We know , so

To get the new value of f, we add Δ y to the old value of f:

The line below has a slope of 2. Find the indicated number.

We use the same formula as before. The slope is 2 and Δ x = -0.5, so

Δ y = 2(-0.5) = -1.

Then

y_new = y_old + (-1) = 4 - 1 = 3.

This makes sense, if you think about it. Having a slope of 2 means for every step of size 1 that x moves right, y should move up 2. This means if x moves to the right by 0.5, y should move up by 1. So the value of y when x = 4 should be one more than the value of y when x = 3, which is what happens.