Question Statement

Draw the graph of the function $y=x^2+4$ and then sketch the graphs of the following functions using the principles of translation. Verify the results using a graphical calculator:

(a) $y=(x^2+4)-3$

(b) $y=(x^2+4)+3$

(c) $y=(x-3{)}^2+4$

(d) $y=(x+3{)}^2+4$

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Background and Explanation

Function translation involves shifting a graph without changing its shape. A vertical translation $y=f(x)+k$ moves the graph up or down, while a horizontal translation $y=f(x-h)$ moves the graph left or right.

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Solution

To sketch these functions, we first establish the coordinates for the base function $y=x^2+4$ and then apply the specific translations for each part.

Table of Values

The following table calculates the $y$-values for the base function and its translated versions across a range of $x$ values:

Part (a): Vertical Translation Down

The function $y=(x^2+4)-3$ simplifies to $y=x^2+1$. By subtracting 3 from the entire function, every point on the original graph $y=x^2+4$ is shifted downward by 3 units. For example, the vertex moves from $(0,4)$ to $(0,1)$.

Part (b): Vertical Translation Up

The function $y=(x^2+4)+3$ simplifies to $y=x^2+7$. By adding 3 to the entire function, every point on the original graph is shifted upward by 3 units. The vertex moves from $(0,4)$ to $(0,7)$.

Part (c): Horizontal Translation Right

The function $y=(x-3{)}^2+4$ represents a horizontal shift. When we replace $x$ with $(x-3)$, the graph moves 3 units to the right. Notice in the table that the $y$-values for $y=x^2+4$ now appear 3 units later in the $x$ sequence. The vertex moves from $(0,4)$ to $(3,4)$.

Part (d): Horizontal Translation Left

The function $y=(x+3{)}^2+4$ represents a horizontal shift. When we replace $x$ with $(x+3)$, the graph moves 3 units to the left. The vertex moves from $(0,4)$ to $(-3,4)$.

Graphical Representation

Below are the visual representations of these translations:

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Key Formulas or Methods Used

• Vertical Shift: $y=f(x)+k$
  • If $k>0$, shift up.
  • If $k<0$, shift down.
• Horizontal Shift: $y=f(x-h)$
  • If $h>0$ (e.g., $(x-3)$), shift right.
  • If $h<0$ (e.g., $(x+3)$), shift left.

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Summary of Steps

1. Identify the Base Function: Determine the parent graph, which is $y=x^2+4$.
2. Calculate Key Points: Create a table of values for $x$ and $y$ to understand the shape and position of the base parabola.
3. Apply Vertical Shifts: For parts (a) and (b), adjust the $y$-coordinates by adding or subtracting the constant.
4. Apply Horizontal Shifts: For parts (c) and (d), adjust the $x$-input within the squared term to shift the graph left or right.
5. Sketch and Verify: Draw the new parabolas based on the shifted vertices and points, then confirm with a graphing utility.