Question

1. reinforce write a function rule to match each of the transformations to the graph of

$y = \sqrt { x }$ described below.
a. the parent function has been shifted left 5 units and up 1 unit.
b. the parent function has been reflected across the x-axis and shifted down 3 units.
c. the parent function has been vertically stretched by a factor of 3, shifted right 2
units, and shifted up 4 units.

Explanation:

Step1: Recall transformation rules

For a function \(y = f(x)\), shifting left \(h\) units gives \(y=f(x + h)\), shifting right \(h\) units gives \(y = f(x - h)\), shifting up \(k\) units gives \(y=f(x)+k\), shifting down \(k\) units gives \(y = f(x)-k\), vertical stretch by a factor \(a\) gives \(y = a\cdot f(x)\), and reflection across the \(x\) - axis gives \(y=-f(x)\).

Step2: Solve part (a)

The parent function \(y = \sqrt{x}\). Shifting left \(5\) units: replace \(x\) with \(x + 5\) (so \(y=\sqrt{x + 5}\)), then shifting up \(1\) unit: add \(1\) to the function. The function is \(y=\sqrt{x + 5}+1\).

Step3: Solve part (b)

The parent function \(y=\sqrt{x}\). Reflection across the \(x\) - axis: \(y =-\sqrt{x}\). Shifting down \(3\) units: subtract \(3\) from the function. The function is \(y=-\sqrt{x}-3\).

Step4: Solve part (c)

The parent function \(y=\sqrt{x}\). Vertical stretch by a factor of \(3\): \(y = 3\sqrt{x}\). Shifting right \(2\) units: replace \(x\) with \(x - 2\) (so \(y = 3\sqrt{x - 2}\)). Shifting up \(4\) units: add \(4\) to the function. The function is \(y=3\sqrt{x - 2}+4\).

Answer:

a. \(y=\sqrt{x + 5}+1\)
b. \(y=-\sqrt{x}-3\)
c. \(y=3\sqrt{x - 2}+4\)