Write the following as piecewise functions

Question

Write the following as piecewise functions: a) 𝑦 = |5𝑥 + 2|  b) 𝑦 = |3 − 4𝑥| + 8

Answer 1

Remember that the shape of an absolute value function is a 'V'. The vertex of this 'V' shape is the point where the function y takes its lowest value. In order to complete this problem there is one key that will help: Adding and subtracting numbers inside the absolute value bars will shift the 'V' to the left and right, while numbers outside the bars will raise and lower the 'V'.

With that said, let's find the lowest point for a). The graph for this function has not been shifted up or down from the parent function (i.e. y = |x|). Therefore, we need to solve 5x + 2 = 0. In this case, subtracting two from both sides and then dividing by 5, we find that x = -2/5, which is -0.4. Now we know that at the x = -2/5 is where the function splits off into two directions. To the right of this value, in other words x > -2/5, the graph looks like a linear function with a positive slope. The easy way to find this slope is to look at the coefficient in the original function. Another way to calculate it is to use the slope formula. At x = 0, y is 2. At x = 5, y is 27. Plugging into the slope formula, we get:

m = (27-2)/(5-0) = 25/5 = 5.

We know that a linear function takes the form y = mx + b. Now we just need the y-intercept, which is the point where x = 0. To do this, we'll need a point we know is on the line. How about the one we already calculated? (-2/5, 0) is the vertex, so we know it goes through both sides of our piecewise function.

y = mx + b

0 = 5(-2/5) + b

0 = -2 + b

b = 2

So, the equation for the right side of our function is y = 5x + 2 for all x values greater than -2/5. We follow the same procedure for the left side of the equation to complete the piecewise function:

y = { 5x + 2, for all x > -2/5}

{-5x -2, for all x < -2/5}

We could do the same process for b) as well, which would give us:

y = {4x + 5, for all x > 3/4}

{-4x + 11, for all x < 3/4}

Answer 2

Recall that l A l = A if A ≥ 0 and l A l = -A when A < 0.

5x + 2 = 0 when x = -2/5.

5x + 2 > 0 when x > -2/5 and 5x. + 2 < 0 when x < -2/5

So, y = 5x + 2 if x ≥ -2/5 and y = -5x - 2 if x < -2/5

The other problem can be done in a similar manner.