Question

1. $t(p) = 7|p - 3| - 4$

vertex: \t\t\topens:
narrower / wider / same (as parent function)

Explanation:

Step1: Recall the vertex form of absolute value function

The parent function of absolute value is \(y = |x|\), and its vertex form is \(y = a|x - h| + k\), where \((h, k)\) is the vertex, \(a\) determines the direction and the width.
For the function \(t(p)=7|p - 3|-4\), comparing with \(y = a|x - h| + k\), we have \(h = 3\), \(k=- 4\), \(a = 7\).

Step2: Determine the vertex

From the vertex form, the vertex \((h,k)\) is \((3,-4)\).

Step3: Determine the direction it opens

Since \(a = 7>0\), the graph opens upwards.

Step4: Determine if it's narrower, wider or same as parent function

The parent function \(y = |x|\) has \(a = 1\). When \(|a|>1\), the graph is narrower than the parent function; when \(0<|a|<1\), the graph is wider. Here \(|7| = 7>1\), so it is narrower.

Answer:

Vertex: \((3, - 4)\)
Opens: Upwards
Narrower / Wider / Same: Narrower