Math · College algebra · Worked example

Reflect and shift an absolute value graph

Describe the graph of g(x) = −|x + 1| + 3 and find its vertex and x-intercepts.

g (x) = - ∣ x + 1 ∣ + 3

Read a, h and k

g(x) = −f(x − (−1)) + 3 with f(x) = |x|: a = −1, h = −1 and k = 3.

g (x) = - f (x - (- 1)) + 3

The V of y = |x| flips to open downward, shifts left 1 and rises 3, so its vertex moves from (0, 0) to (−1, 3).

Set g(x) = 0: |x + 1| = 3, so x + 1 = 3 or x + 1 = −3. The intercepts are x = 2 and x = −4, three units either side of the vertex.

∣ x + 1 ∣ = 3 ⟹ x = 2 or x = - 4

\begin{array}{c} ∣ x + 1 ∣ = 3 \\ x = 2 or x = - 4 \end{array}

The vertex and both intercepts satisfy the formula.

\begin{array}{l} g (x) = - ∣ x + 1 ∣ + 3 \\ g (- 1) = 3 \\ g (2) = 0 \\ g (- 4) = 0 \end{array}

Vertex (−1, 3), opening downward; x-intercepts x = −4 and x = 2.

Give the vertex and the y-intercept of k(x) = 2|x − 1| − 4.

Show the answer and explanation

k is |x| stretched by 2, shifted right 1 and down 4, so the vertex is (1, −4). At x = 0, k(0) = 2|−1| − 4 = −2.

\begin{array}{l} k (x) = 2 ∣ x - 1 ∣ - 4 \\ k (1) = - 4 \\ k (0) = - 2 \end{array}

In Graph, the V has slopes 1 and −1 on either side of its vertex. Change the leading −1 to −2 and the intercepts move in to x = 0.5 and x = −2.5.

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