Problem 8 - 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. $$ h(x)=(x-1)^{2}+2 $$

Step-by-Step Solution

Answer

The graph of the equation \(h(x)=(x-1)^{2}+2\) is a parabola that opens upwards with the vertex at the point (1,2). This is obtained by shifting the graph of \(f(x)=x^{2}\) one unit to the right and two units up.

Step 1: Graph the standard quadratic function \(f(x)=x^{2}\)

Start by graphing the standard quadratic function \(f(x)=x^{2}\). This is a parabola opening upwards with the vertex at the origin (0,0). The graph is symmetric with respect to the y-axis.

Step 2: Understand the transformations

The function \(h(x)=(x-1)^{2}+2\) represents the function \(f(x)=x^{2}\) shifted 1 unit to the right and 2 units up. This is because the -1 inside the parenthesis with \(x\) moves the graph to the right and the +2 outside affects vertical shift, moving the graph upwards.

Step 3: Apply the transformations and graph \(h(x)\)

Now apply these transformations to the graph of \(f(x)\). Move the vertex of the parabola one unit to the right which will place it at (1,0). Then, shift the graph two units up, taking the vertex to (1,2). The graph of \(h(x)\) is still a parabola opening upwards, but now with the vertex at (1,2).

Key Concepts

Graph Transformations

In mathematics, graph transformations are changes made to the graph of a function that alter its appearance or position. Understanding these transformations helps to sketch functions more efficiently. For quadratic functions, these transformations might include shifts, stretches, compressions, and reflections. When dealing with quadratic functions like

• **Vertical Shifts**: Adding or subtracting a number outside the squared term moves the graph up or down. For example, adding 2 to the function results in an upward shift by 2 units.
• **Horizontal Shifts**: Adjusting the value inside the parentheses with the squared variable affects its horizontal placement. Subtracting a value shifts the graph rightward, as seen when changing from \(f(x) = x^2\) to \(f(x) = (x-1)^2\).
• **Reflections and Stretches**: While not part of this exercise, it's helpful to remember reflection across the x-axis or stretching in any direction via multiplication by a factor.

These transformations allow us to understand and manipulate the function to obtain various forms without needing to re-calculate the entire function's characteristics each time.

Vertex Form of a Quadratic

The vertex form of a quadratic function is given by \(y = a(x-h)^2 + k\). It's a powerful way to write quadratics because it clearly points out the vertex, (h, k), of the parabola. This form makes it easier to graph, revealing the direction and position of the parabola in an instant.

• For the function \(h(x)=(x-1)^2+2\), the vertex is (1,2), which is straightforward from the expression.
• **Vertex Identification**: Looking at \((x-1)^2\), we see a horizontal shift 1 unit to the right from the standard position \(x^2\), with \(1\) determining this horizontal move.
• **Vertical Position**: With \(+2\) outside the square, we know the vertex shifts 2 units up.

Understanding these different components of the vertex form helps simplify finding a parabola’s highest or lowest point without additional calculations.

Parabolic Shifts

Parabolic shifts involve moving the entire graph of a quadratic function in the coordinate plane. These shifts involve horizontal and vertical movements that change the vertex location but retain the shape and orientation of the parabola.

• **Horizontal Shifts**: Seen in terms like \(x-1\), a horizontal shift is determined by the value subtracted (or added) inside the square. This shift doesn't affect the U-shape of the parabola but carries it left or right.
• **Vertical Shifts**: Vertical movements adjust the up or down position of the vertex. For example, in \(h(x) = (x-1)^2 + 2\), adding 2 represents a move up by 2 units.

These shifts are straightforward but provide insight into how a quadratic's graph can be repositioned for specific requirements or starting points. Keeping track of these transformations is crucial when creating graphs or solving any algebraic equations involving quadratics.