Problem 28 - Precalculus : Building Concepts and Connections

Question

Use transformations to graph the quadratic function and find the vertex of the associated parabola. $$h(x)=(x-3)^{2}+2$$

Step-by-Step Solution

Answer

The vertices of the graph of the function \(h(x) = (x-3)^2 +2\) is at point (3, 2).

Step 1: Identify the Transformations

To identify the transformations, subtract 3 from x in the parent function and add 2 to the function value. This changes the equation from y = x² to h(x) = (x - 3)² + 2. This moves the parent function to the right by 3 units and up by 2 units.

Step 2: Plot the Parabola

Start by plotting the vertex (3, 2) since the parabola has been shifted right by 3 units and up by 2 units from the origin. Then, take points on either side of the vertex and find their corresponding function values. This will let you sketch the full parabola.

Step 3: Identify the Vertex

The vertex of the parabola is at point (h, k), where h and k are the numbers in the equation in the form \(h(x) = (x-h)^2 + k\). With this function, the vertex is at point (3, 2).

Key Concepts

Quadratic Functions

Quadratic functions are a central concept in algebra and appear in the form of \( f(x) = ax^2 + bx + c \). The key feature of a quadratic function is its highest degree term, which is the square of the variable, hence 'quadratic'. These functions form parabolas when graphed. The equation form can be manipulated through transformations to better understand or graph them.

In the standard form \( f(x) = ax^2 + bx + c \), when transforming to vertex form \( h(x) = a(x-h)^2 + k \), the vertex is directly visible as \((h, k)\). This is particularly useful for graphing and understanding how the graph shifts or changes its shape due to different values of \(a\), \(h\), and \(k\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.
• The vertex \( (h, k) \) is derived directly from the transformed equation.

Understanding these transformations helps in graphing quadratics more efficiently.

Parabolas

Parabolas are the U-shaped graphs that are formed by quadratic functions. They have a symmetrical property, key characteristics like vertex, axis of symmetry, and can often open upwards or downwards depending on the coefficients.

The vertex of a parabola, in the context of the vertex form \( h(x) = a(x-h)^2 + k \), is visually the "turning point". It is either the highest or the lowest point on the graph depending on the sign of \(a\). The axis of symmetry is a vertical line that passes through the vertex, with the equation \(x = h\).

When working with parabolas, remember:

• The vertex determines the "peak" or "valley" of the parabola.
• The axis of symmetry provides a line over which each half of the parabola is a mirror image.
• Points equidistant from the axis of symmetry have the same y-value.

This makes sketching or understanding the nature of quadratic functions simpler.

Graphing Transformations

Graphing transformations involve shifting or scaling the graph of a function. In quadratic functions, these transformations help us easily graph and understand changes in the function's graph.

For the function \( h(x)=(x-3)^2+2 \):

• The \(x-3\) indicates a horizontal shift to the right by 3 units.
• The \(+2\) indicates a vertical shift upwards by 2 units.

These transformations are what alter the position of the vertex from \((0, 0)\) in the basic \(y=x^2\) graph to \((3, 2)\) in the function \(h(x)\). These shifts make it easy to determine the new position of the vertex as well as any other points of interest to sketch the graph.

Understanding these transformations ensures that when given any quadratic function in vertex form, you can accurately sketch its graph and identify primary features like the vertex and axis of symmetry.