Why does multiplying the input of a linear function change only the slope while multiplying the output changes both the slope and the
y-intercept?

Step-by-step explanation:

If we have a linear function and we multiply the input of the function, we aren't really changing the values of the function, we're changing the rate at which those values are reached.

E.g. if you look at the attached picture, we have and , which is just . The value when the red line is x = 1 is half that of the value of the blue line when x = 1.

The values of the f(2x) line are reaching the values of f(x) twice as fast, meaning it has a slope of twice as much. However, it still grows through the same intercept.

If you're interested in the calculus behind this, look up the chain rule of derivatives.

However, when you start changing the output of the function, you're changing the entire equation. It's not the same line anymore.

step-by-step explanation:

here's what happens:

1) the graph of f   = x^2 is shifted 2 units to the right.

2) the resulting graph is reflected in the x-axis, so that the parabola now opens down.

3) this final result is transformed up by 5 units.

costs the what? what is the cost of the pants?

step-by-step explanation:

if it is $10, then the price is 14.

take 40% of the cost, and then add the cost.

40% of 10 = 4

4+10 = 14.