Problem 58 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ h(x)--(x-1)^{2} $$

Step-by-Step Solution

Answer

The graph of function \(h(x) = (x - 1)^{2}\) is a parabola opening upward, with its vertex shifted 1 unit to the right from the origin, compared to the graph of \(f(x) = x^{2}\).

Step 1: Start with the Standard Quadratic Function

The function to start with is \(f(x) = x^{2}\). Its graph is a parabola opening upward, with the vertex at the origin.

Step 2: Understand the Transformation Needed

The function \(h(x) = (x - 1)^{2}\) is derived from \(f(x) = x^{2}\) by replacing \(x\) with \((x - 1)\). This is a horizontal shift of the graph of \(f\) to the right by 1 unit.

Step 3: Apply the Transformation

The graph of \(h(x) = (x - 1)^{2}\) is exactly the same as that of \(f(x) = x^{2}\), but shifted 1 unit to the right.

Step 4: Plot the New Function

To graph \(h(x) = (x - 1)^{2}\), start at the vertex of the parabola of \(f\), which is at the origin, and move it 1 unit to the right to get the vertex of \(h\). The graph still remains a parabola opening upward.

Key Concepts

Standard Quadratic Function

The standard quadratic function, represented as f(x) = x^2, is a foundational element in understanding more complex quadratic equations. The graph of this function creates a shape known as a parabola. A parabola has a unique characteristic of being symmetrical, with the axis of symmetry passing through its highest or lowest point, called the vertex.

For the standard quadratic function, the vertex is at the origin point (0,0), which also serves as the location of the minimum value for graphs opening upwards. The function is simple to graph by plotting a set of points that satisfy the equation y = x^2 and drawing a curve through these points to illustrate the upward opening parabola.

Parabola Transformations

Transformations of parabolas are modifications that alter the graph's size, direction, position, or orientation. There are four primary types of transformations: translation, reflection, stretch, and compression.

Translation

A translation involves shifting the graph horizontally or vertically without changing its shape or orientation. It's akin to sliding a picture across a wall. When we talk about translating a function horizontally or vertically, we use the form h(x) = a(x-h)^2 + k. In this notation, h represents the horizontal shift and k the vertical shift.

Reflection

A reflection is a flipping of the graph over a line, often the x-axis or y-axis. This is what happens when you see your mirror image.

Stretch and Compression

These transformations either widen (compression) or elongate (stretch) the graph of the parabola by a factor. If the stretching or compressing factor (a) is greater than 1, the graph is steeper; if between 0 and 1, the graph is wider.

Horizontal Shift

A horizontal shift is a type of translation that moves the graph of a function left or right along the x-axis.

When graphing quadratic functions, a horizontal shift can be observed when we replace x with (x-h) where h is a constant. The magnitude of the shift is the absolute value of h, and the direction of the shift depends on the sign. A positive h leads to shifting to the right, while a negative h moves it to the left.

The horizontal shift is commonly misunderstood, which is why it's important to remember that the function h(x) = (x-1)^2 represents a shift to the right by 1 unit. The mistake often made is thinking that since the expression is x-1, the graph moves left, but it's the opposite. To properly graph a function with a horizontal shift, one should begin plotting at the new translated vertex and then sketch the parabola from there.