Problem 56 - Elementary and Intermediate Algebra

Question

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function \(g\) using translations and/or a reflection. See Examples 7 and \(8 .\) $$ g(x)=(x-3)^{2}-2 $$

Step-by-Step Solution

Answer

Graph the parabola opening upwards with vertex at (3, -2).

Step 1: Identify the base function

The given function is \(g(x) = (x-3)^2 - 2\). Notice that this function is a transformation of the base quadratic function \(f(x) = x^2\). We’ll focus on this base function for our transformations.

Step 2: Sketch the base function

First, we sketch the basic quadratic function \(f(x) = x^2\) which is a parabola opening upwards with its vertex at the origin (0, 0).

Step 3: Apply horizontal translation

The function \((x-3)^2\) indicates a horizontal translation of the basic quadratic parabola by 3 units to the right. This moves the vertex from (0, 0) to (3, 0).

Step 4: Apply vertical translation

The term \(-2\) in \((x-3)^2 - 2\) indicates a vertical translation downward by 2 units. This moves our translated vertex from (3, 0) to (3, -2).

Step 5: Combine transformations in a sketch

The final graph of \(g(x) = (x-3)^2 - 2\) is the parabola \(f(x) = x^2\), moved 3 units to the right and 2 units down. So, the vertex of this parabola is at (3, -2). The graph maintains its shape and direction, opening upwards.

Key Concepts

Quadratic Functions

Quadratic functions are fundamental in understanding various transformations due to their simple yet flexible structure. These functions are often written in the form: \(f(x) = ax^2 + bx + c\). The most basic form is \(f(x) = x^2\), a parabolic shape that opens upwards with its vertex at the origin, (0, 0). This vertex is a critical point since it indicates the lowest point on the graph, given that the parabola opens upwards. The symmetry and simplicity of quadratic functions make them an ideal starting point for graph transformations.

• *Quadratic Formula:* In standard form, the vertex is located at \(x = -\frac{b}{2a}\).
• *Direction of Opening:* The sign of \(a\) determines if the graph opens upwards (positive) or downwards (negative).

Understanding the structure of quadratic functions allows for easy application of transformations that shift and modify the graph’s position.

Horizontal Translation

Horizontal translation refers to the shifting of a graph along the x-axis. When we alter the x-variable inside a function, we execute a horizontal shift. For quadratic functions, an expression like \(g(x) = (x - h)^2\) indicates a horizontal movement of \(h\) units. But remember, the direction might seem counterintuitive; \((x - 3)^2\) moves the graph 3 units to the right, even though there is a negative sign.

• *Positive \(h\):* Move left.
• *Negative \(h\):* Move right.

This translation does not change the shape or the direction in which the parabola opens. It only reposition the parabola on the x-axis, impacting the location of the vertex from the origin \(0, 0\) to \(h, 0\).

Implications of Horizontal Shifts

Knowing how horizontal shifts affect the graph helps in precisely sketching the function. This step is crucial before any vertical translations.

Vertical Translation

Vertical translation involves shifting the graph up or down along the y-axis. This happens when there is a constant added or subtracted to the function as a whole. For a function like \(g(x) = x^2 - k\), the value \(k\) determines this translation, moving the graph upwards if \(k > 0\) or downwards if \(k < 0\).

• *Positive \(k\):* Graph moves upwards.
• *Negative \(k\):* Graph moves downwards.

The vertical translation modifies where the vertex of the parabola settles after any horizontal translation. In terms of our function \(g(x) = (x-3)^2 - 2\), the shift down by 2 units moved the vertex from \(3, 0\) to \(3, -2\).

Effect on Graph Shape

Vertical shifts do not alter the width or the direction of the parabola. They simply adjust the elevation of the entire graph, dictating where the graph narrowly touches the y-axis, aiding comprehensive graph representation.