Question

practise

1. for each function, state the values of \\(h\\) and \\(k\\), the parameters that represent the horizontal and vertical translations applied to \\(y = f(x)\\).

a) \\(y - 5 = f(x)\\)
b) \\(y = f(x) - 4\\)
c) \\(y = f(x + 1)\\)
d) \\(y + 3 = f(x - 7)\\)

Explanation:

Define the standard transformation form

We analyze the general form of a translated function.
The standard form representing horizontal and vertical translations applied to \(y = f(x)\) is:

$$y - k = f(x - h)$$

where:

• \(h\) represents the horizontal translation.
• \(k\) represents the vertical translation.

We will apply Function Transformations to identify these parameters for each given equation.

Analyze equation a

We rewrite the equation to match the standard form.
Given:

$$y - 5 = f(x)$$

Comparing this to \(y - k = f(x - h)\):

• The left side is \(y - 5\), so \(k = 5\).
• The right side is \(f(x)\), which is \(f(x - 0)\), so \(h = 0\).

Analyze equation b

We rewrite the equation to match the standard form.
Given:

$$y = f(x) - 4$$

Rearranging by adding \(4\) to both sides:

$$y + 4 = f(x)$$

Which can be written as:

$$y - (-4) = f(x - 0)$$

Comparing to \(y - k = f(x - h)\):

• \(h = 0\)
• \(k = -4\)

Analyze equation c

We rewrite the equation to match the standard form.
Given:

$$y = f(x + 1)$$

Which can be written as:

$$y - 0 = f(x - (-1))$$

Comparing to \(y - k = f(x - h)\):

• \(h = -1\)
• \(k = 0\)

Analyze equation d

We rewrite the equation to match the standard form.
Given:

$$y + 3 = f(x - 7)$$

Which can be written as:

$$y - (-3) = f(x - 7)$$

Comparing to \(y - k = f(x - h)\):

• \(h = 7\)
• \(k = -3\)

Answer:

Question 1

a) \(h = 0\), \(k = 5\)

b) \(h = 0\), \(k = -4\)

c) \(h = -1\), \(k = 0\)

d) \(h = 7\), \(k = -3\)