Problem 23 - Algebra and Trigonometry

Question

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=x^{2}, \quad g(x)=(x+2)^{2}\) (b) \(f(x)=x^{2}, \quad g(x)=x^{2}+2\)

Step-by-Step Solution

Answer

(a) Shift left 2 units; (b) Shift up 2 units.

Step 1: Understand the Function f(x)

The given function is \(f(x) = x^2\), which is a standard quadratic function. The graph of \(f(x)\) is a parabola that opens upwards with its vertex at the origin (0,0).

Step 2: Analyze Part (a) Transformation

In part (a), the function \(g(x) = (x+2)^2\) is provided. This represents a horizontal transformation. Specifically, adding 2 inside the parenthesis shifts the graph of \(f(x)\) to the left by 2 units. Thus, the vertex of the parabola moves from (0,0) to (-2,0).

Step 3: Analyze Part (b) Transformation

In part (b), the function \(g(x) = x^2 + 2\) represents a vertical transformation. Adding 2 outside the parenthesis shifts the graph of \(f(x)\) upwards by 2 units. Therefore, the vertex of the parabola moves from (0,0) to (0,2).

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. The graph of a quadratic function is a parabola. These parabolas can open upwards or downwards.

• If \( a > 0 \), the parabola opens upwards, forming a "U" shape.
• If \( a < 0 \), it opens downwards, forming an inverted "U" shape.

The most basic quadratic function is \( f(x) = x^2 \), which is known as the parent quadratic function. Its graph is a parabola centered at the origin (0,0). This vertex is the point at the peak or the trough of the parabola, depending on the direction it's opening. Watching the transformation of this function can help understand more complex quadratic functions. Recognizing shifts and transformations is key to mastering quadratic functions.

Horizontal Shift

A horizontal shift refers to moving the graph of a function left or right along the x-axis. For a quadratic function \( f(x) = (x-h)^2 \), the term \( h \) within the parenthesis indicates a horizontal shift.

• When you add a positive number \( h \) inside the parenthesis \((x + h)^2\), it shifts the graph to the left by \( h \) units.
• Conversely, subtracting a number \( h \) \((x - h)^2\), shifts it to the right by \( h \) units.

In our example, \( g(x) = (x+2)^2 \) results from shifting the basic quadratic function \( f(x) = x^2 \) to the left by 2 units. As such, the vertex moves from (0,0) to (-2,0). Always recall that the direction might seem counterintuitive since adding results in a left shift and subtracting results in a right shift. Visualizing these shifts on a graph can clarify this concept further.

Vertical Shift

Vertical shift involves moving the graph of a function up or down along the y-axis. In the case of quadratic functions, a vertical shift can be seen by adding or subtracting a constant \( k \) to or from the function.

• Adding a positive number \( k \) to \( f(x) \), as in \( g(x) = x^2 + k \), shifts the graph upwards by \( k \) units.
• Subtracting a positive number \( k \) results in a downward shift.

For our second scenario, \( g(x) = x^2 + 2 \), indicates the graph of \( f(x) = x^2 \) is shifted up by 2 units. Thus, the vertex elevation alters from (0,0) to (0,2). Vertical shifts are somewhat intuitive, as moving up translates to addition and moving down translates to subtraction. Observing how these adjustments work in combination with horizontal shifts will deepen your comprehension of quadratic transformations. Understanding and visualizing both shifts together helps in graphing complex polynomial functions.