Problem 70 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to solve the problem. Graph \(f(x)=(x-4.5)^{2}\) and \(g(x)=x^{2}+4.5 .\) Describe each graph in terms of transformations of the graph of \(h(x)=x^{2}\).

Step-by-Step Solution

Answer

The function \(f(x) = (x - 4.5)^2\) is the graph of \(h(x) = x^2\) shifted 4.5 units to the right, and \(g(x) = x^2 + 4.5\) is the graph of \(h(x) = x^2\) shifted 4.5 units upward.

Step 1: Graph f(x)

Graph the function \(f(x) = (x - 4.5)^2\), which is a horizontal transformation of the function \(h(x) = x^2\). This is done by shifting the graph of \(h(x) = x^2\) to the right by 4.5 units.

Step 2: Describe Transformation of f(x)

The graph of \(f(x) = (x - 4.5)^2\) is a shift 4.5 units to the right of the graph of \(h(x) = x^2\). This is a horizontal shift - in other words, the transformation has shifted the function horizontally.

Step 3: Graph g(x)

Graph the function \(g(x) = x^2 + 4.5\), which is a vertical transformation of the function \(h(x) = x^2\). This is done by shifting the graph of \(h(x) = x^2\) upwards by 4.5 units.

Step 4: Describe Transformation of g(x)

The graph of \(g(x) = x^2 + 4.5\) is a shift 4.5 units upward of the graph of \(h(x) = x^2\). This is a vertical shift - in other words, the transformation has shifted the function vertically.

Key Concepts

Horizontal Shift

Understanding the horizontal shift in graph transformations is essential to mastering precalculus. A horizontal shift occurs when every point on the graph of a function moves a specific number of units to the left or to the right along the x-axis.

For example, if we take the function h(x) = x^2 and modify it to f(x) = (x - 4.5)^2, we've applied a horizontal shift. This shift means that for any x-value, the corresponding y-value from the parent function h(x) will now occur at an x-value that is 4.5 units to the right.

It's important to remember that the sign inside the brackets indicates the direction of the shift. If it’s (x - a), the graph moves right by 'a' units, and if it’s (x + a), the graph moves left by 'a' units. So, in our example, the graph of h(x) is shifted 4.5 units to the right to create the graph of f(x).

Vertical Shift

The vertical shift in graph transformations can be easily grasped by picturing the entire graph moving up or down. Unlike horizontal shifts, where we move along the x-axis, vertical shifts occur along the y-axis.

Let's consider the function g(x) = x^2 + 4.5. Here, the '+4.5' outside the squared term adds to every y-value of the parent function h(x) = x^2. Consequently, every point on the graph moves up by 4.5 units, giving us a vertical shift.

This kind of shift is straightforward: if the constant added to x^2 is positive, the shift is upwards, and if it's negative, the graph moves downwards. In summary, g(x) = x^2 + 4.5 results in a vertical uplift of the graph of h(x) by 4.5 units.

Graphing Quadratic Functions

Graphing quadratic functions forms the bedrock of many concepts in precalculus, especially transformations. A quadratic function typically has the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, and its shape and position can be influenced by each of the coefficients a, b, and c.

When a is positive, the parabola opens upward, inversely, it opens downward when a is negative. The coefficient a also affects the 'width' of the parabola; larger values create a narrower shape, while smaller values widen it.

The c value will move the graph up or down (vertical shift), and the b value, in conjunction with a, determines the location of the vertex and the parabola's axis of symmetry. It's these transformations that allow us to modify the base function h(x) = x^2 to fit various scenarios and equations.

Transformations of Functions

Transformations of functions are operations that modify the graph of a function in various ways. Common transformations include translations, reflections, stretches, and compressions.

A translation moves the graph horizontally or vertically, as we've explored with the shifts. In addition to these, reflecting the function across the x-axis or y-axis can invert its graph. A stretch or compression involves widening or narrowing the graph, respectively. These are often caused by multiplying the function by a factor.

• If the multiplication is outside the main function, e.g., 2f(x), it's a vertical stretch/compression.
• If it's within the function, e.g., f(2x), it's a horizontal stretch/compression.

Understanding these transformations allows you to predict and manipulate the shape and positioning of a function's graph, helping to solve problems in algebra and calculus with greater ease.