Question

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) \(y=f(x)-4\) \(y=f(x+1)\) d) \(y+3=f(x-7)\) e) \(y=f(x+2)+4\)

Short answer

a) h = 0, k = 5b) h = 0, k = -4c) h = -1, k = 0d) h = 7, k = -3e) h = -2, k = 4

Step-by-step solution

Identify Translations in Exercise a

Given the function: \(y - 5 = f(x)\). To rewrite it in the form \(y = f(x) + k\), we add 5 to both sides: \(y = f(x) + 5\). Here, the parameter \(k\) represents a vertical translation. Therefore, \(k = 5\) and there is no horizontal translation, so \(h = 0\).

Identify Translations in Exercise b

Given the function: \(y = f(x) - 4\). This is already in the form \(y = f(x) + k\) where \(k = -4\). Here, the vertical translation is \(k = -4\) and there is no horizontal translation, so \(h = 0\).

Identify Translations in Exercise c

Given the function: \(y = f(x + 1)\). This is in the form \(y = f(x + h)\) where \(h = -1\) because the horizontal translation is opposite the sign in the function. Therefore, \(h = -1\) and there is no vertical translation so \(k = 0\).

Identify Translations in Exercise d

Given the function: \(y + 3 = f(x - 7)\). To rewrite it in the form \(y = f(x + h) + k\), subtract 3 from both sides: \(y = f(x - 7) - 3\). Here, the horizontal translation is opposite the sign in the function so \(h = 7\). The vertical translation is \(k = -3\).

Identify Translations in Exercise e

Given the function: \(y = f(x + 2) + 4\). This is in the form \(y = f(x + h) + k\) where \(h = -2\) and \(k = 4\). The horizontal translation is \(h = -2\), and the vertical translation is \(k = 4\).

Key concepts

Horizontal Translation

Horizontal translation shifts the graph of a function left or right.
This shift is dictated by the parameter h inside the function, where the function takes the form: y=f(x+h) or y=f(x-h).
If the function is written as y=f(x+h), the graph will shift to the left by h units.
Conversely, if it's y=f(x-h), the graph will shift to the right by h units.
The key thing to remember is that the horizontal translation works opposite to what you might initially think:
- If h is positive in the expression y=f(x+h), the graph moves left.
- If h is negative in the expression y=f(x-h), the graph moves right.
Recognizing horizontal translations helps you understand how the function's graph moves along the x-axis, adjusting its placement but not its shape.

Vertical Translation

Vertical translation shifts the graph of a function up or down.
This shift is controlled by the parameter k which appears added or subtracted from the function as: y=f(x)+k or y=f(x)-k.
If k is positive, the graph of the function moves up by k units.
If k is negative, the graph of the function moves down by k units.
Vertical translations change the y-values of the function's points, making the graph move vertically:
- With y=f(x)+k, if k is positive, move the graph up.
- With y=f(x)-k, if k is negative, move the graph down.
Understanding vertical translations is crucial to see how the function's graph shifts along the y-axis while the shape of the graph remains unchanged.

Parameters

Parameters h and k specify how a function graph translates in the coordinate plane.
Together, these parameters provide guidelines for shifting the graph of y=f(x) horizontally and vertically:
- h influences horizontal translation:
\(\text{y=f(x+h) or y=f(x-h)}\).
- k dictates vertical translation:
\(\text{y=f(x)+k or y=f(x)-k}\).
When examining a function for translations, locate h and k in the transformed function and apply:
- h's effect to move the graph left or right.
- k's effect to move the graph up or down.
This makes it much easier to understand and predict the new position of the function's graph accurately. Utilizing parameters correctly helps in breaking down how any function has been shifted from its original position.