Problem 15 - Algebra and Trigonometry

Question

\(15-18\) Explain how the graph of \(g\) is obtained from the graph of \(f .\) $$ \begin{array}{l}{\text { (a) } f(x)=x^{2}, \quad g(x)=(x+2)^{2}} \\ {\text { (b) } f(x)=x^{2}, \quad g(x)=x^{2}+2}\end{array} $$

Step-by-Step Solution

Answer

For (a), shift 2 units left; for (b), shift 2 units up.

Step 1: Understand the Base Function

The base function for both parts (a) and (b) is \( f(x) = x^2 \), which is a parabola opening upwards with its vertex at the origin (0,0).

Step 2: Analyze Part (a) Transformation

In part (a), we have \( g(x) = (x+2)^2 \). This indicates a horizontal shift. The \( +2 \) inside the parenthesis means the graph of \( f(x) = x^2 \) is shifted 2 units to the left. Thus, the vertex of the parabola moves from (0,0) to (-2,0).

Step 3: Analyze Part (b) Transformation

In part (b), we have \( g(x) = x^2 + 2 \). This represents a vertical shift. The \( +2 \) at the end indicates the graph of \( f(x) = x^2 \) is shifted 2 units upwards. Therefore, the vertex of the parabola moves from (0,0) to (0,2).

Key Concepts

Horizontal Shift

When transforming the graph of a parabola, a horizontal shift involves moving the graph left or right along the x-axis. This transformation is influenced by changes inside the function's parentheses.
To understand horizontal shifts, consider an example like part (a) of the exercise: from the base function \( f(x) = x^2 \) to \( g(x) = (x + 2)^2 \).
The addition of \(+2\) to \(x\) results in the entire graph moving in the opposite direction to what we might initially assume. In this case, it shifts 2 units to the **left**.
Consequently, the vertex of the original function at the point \((0,0)\) is relocated to \((-2,0)\). This movement alters the x-coordinates of all points on the graph while keeping the y-coordinates unchanged.

• The rule is: if the function includes \( (x + a) \), shift the graph left by \(a\) units.
• If the function includes \( (x - a) \), shift the graph right by \(a\) units.

Understanding this concept is crucial, as it helps predict how each point on a graph is shifted horizontally without altering its shape.

Vertical Shift

A vertical shift involves moving the graph of a function up or down along the y-axis. This occurs when a constant is added or subtracted from the entire function.
For instance, in part (b) of the exercise, the transformation of the function from \( f(x) = x^2 \) to \( g(x) = x^2 + 2 \) reflects a vertical shift.
The \(+2\) added to the function signifies a movement of the whole graph 2 units **upwards**.
This alteration changes the y-coordinate of each point on the graph while maintaining the x-coordinate, effectively raising the parabola without transforming its shape or width.

• Adding a positive constant moves the graph upwards by that many units.
• Subtracting a constant moves the graph downwards by that many units.

So, for \( g(x) = x^2 + 2 \), the parabola’s vertex, originally at \( (0,0) \), moves to \( (0,2) \). Recognizing vertical shifts is vital, as they convey how function values increase or decrease uniformly, moving the graph vertically.

Vertex of Parabola

The vertex of a parabola is a pivotal point where it either reaches its highest or lowest point, depending on the orientation. Understanding movements and transformations that affect the vertex is essential.
For the base function \( f(x) = x^2 \), the vertex is at the origin \( (0,0) \).
This vertex shifts due to transformations applied to the function.

Impact of Horizontal and Vertical Shifts

Horizontal and vertical shifts are transformations that relocate the vertex:
- In a horizontal shift, as seen in part (a) with \( g(x) = (x + 2)^2 \), the vertex moves horizontally to \( (-2,0) \).
- In a vertical shift, illustrated in part (b) with \( g(x) = x^2 + 2 \), the vertex moves vertically to \( (0,2) \).

Determining the Vertex

Use the formula \( (h,k) \), where \( h \) and \( k \) result from the shifts:
- In both horizontal and vertical transformations, \( h \) is obtained by solving \( x + a = 0 \), leading to \( h = -a \) for horizontal shifts.
- The value of \( k \) comes directly from added constants to the function, defining the new y-coordinate.
Recognizing how these transformations influence the vertex helps accurately graph and understand parabolas.