Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(h(x)=x^{2}-2 x+1\)

Step-by-Step Solution

Answer

The vertex of the parabola for the function \(h(x) = x^{2} - 2x + 1\) is at (1, 0). The graph of the function is a parabola that opens upwards with the vertex at (1, 0) and y-intercept at (0,1).

Step 1: Express the Function in Vertex Form

The given quadratic function is \(h(x) = x^{2} - 2x + 1\). The vertex form of a quadratic function is \(h(x) = a(x-h)^2 + k\), where (h, k) is the vertex of the parabola. To convert our function into this form, we adjust the square term and the linear term, so that we have \(h(x) = (x-1)^2 + 0\).

Step 2: Identify the vertex

From the vertex form \(h(x) = (x-1)^2 + 0\), we can see the vertex (h, k) of the parabola is at the point (1, 0).

Step 3: Graph the quadratic function

To graph \(h(x) = (x-1)^2 + 0\), first plot the vertex point (1,0) on a graph. The parabola opens upwards because the coefficient of \(x^2\) is positive. The y-intercept is \(y = h(0) = 1\), plot this point and draw a parabola through these points.

Step 4: Verify the graph using graphing utility

Use a graphing utility software or tool to plot the function \(h(x) = x^{2} - 2x + 1\). The graph should match the one we drew in the previous step.

Key Concepts

Vertex Form

In mathematics, the vertex form of a quadratic function is a very useful way to represent parabolas. It takes the form \( h(x) = a(x-h)^2 + k \). The elements \( h \) and \( k \) represent the vertex of the parabola, which is a crucial point indicating the parabola's highest or lowest point depending on its orientation. Transforming a standard quadratic equation like \( h(x) = x^2 - 2x + 1 \) into this vertex form involves completing the square. This allows us to easily identify important characteristics of the parabola such as its vertex, direction, and width.

For the exercise at hand, rewriting \( h(x) = x^2 - 2x + 1 \) in vertex form gives \( h(x) = (x-1)^2 + 0 \), clearly showing the vertex at \( (1, 0) \). Understanding the vertex form allows for a straightforward analysis of the quadratic function, enabling quick identification of the vertex and aiding in graphing.

Parabola

A parabola is a symmetrical, U-shaped curve that is a common graph for quadratic functions. The orientation of a parabola is defined by the coefficient \( a \) in the function's vertex form \( h(x) = a(x-h)^2 + k \). If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downward. In our exercise, the given function \( x^2 - 2x + 1 \) has its parabola open upwards because the coefficient of the \( x^2 \) term is positive.

Understanding the shape and orientation of a parabola helps predict how the graph will look and aids in plotting. The vertex plays a significant role since it is the point where the parabola changes direction. Every parabola is symmetric around its vertical axis through the vertex, which means knowing the vertex helps to graph the parabola more accurately.

Graphing Utility

A graphing utility is an essential tool in mathematics for visualizing and analyzing the behavior of functions. These utilities, which can be software or calculators, allow you to input the function and see its graph instantly. This is particularly useful for checking your manual graph against a more precise one created digitally.

For the quadratic function \( h(x) = x^2 - 2x + 1 \), using such a utility can verify the manual graph by accurately plotting the parabola with the vertex at \( (1, 0) \) and ensuring it opens upwards. Additionally, graphing utilities can zoom in on specific parts of the graph, measure distances, and even suggest potential symmetries or intercepts, providing a deeper understanding of the graph's characteristics.

Vertex Identification

Identifying the vertex of a parabola is vital when analyzing quadratic functions. The vertex is the point \( (h, k) \) in the function's vertex form \( h(x) = a(x-h)^2 + k \). It represents the maximum or minimum point of the parabola, where it changes direction. In the standard form, converting \( h(x) = x^2 - 2x + 1 \) into the vertex form \( (x-1)^2 + 0 \) helps identify that the vertex is located at \( (1, 0) \).

Recognizing the vertex enables important insights into the function's graphical representation and properties. Knowing this point, you can easily determine the line of symmetry and predict how the graph behaves on either side of it. It also assists in quickly sketching the parabola, especially in determining other attributes such as focus and directrix, although those might typically require further information.