Problem 13 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(h(x)=16-x^{2}\)

Step-by-Step Solution

Answer

Plot the points and draw a downward-opening parabola with vertex at (0, 16).

Step 1: Choose Values for x

Begin by selecting a range of values for \(x\). It is often helpful to choose both positive and negative values. For this exercise, let's choose values around the vertex of the parabola: \(x = -4, -3, -2, -1, 0, 1, 2, 3, 4\).

Step 2: Calculate Corresponding y-values

Using the chosen \(x\)-values, compute corresponding \(y = h(x)\) values using the function \(h(x) = 16 - x^2\). For example, if \(x = -4\), then \(h(-4) = 16 - (-4)^2 = 16 - 16 = 0\). Continue calculating for the other \(x\)-values.

Step 3: Create a Table

Organize the calculated values into a table to easily see the pairs \((x, y)\):| \(x\) | \(h(x)\) ||------|--------|| -4 | 0 || -3 | 7 || -2 | 12 || -1 | 15 || 0 | 16 || 1 | 15 || 2 | 12 || 3 | 7 || 4 | 0 |

Step 4: Plot the Points on a Graph

Using the table, plot each point \((x, h(x))\) on a coordinate grid. For example, plot \((-4, 0), (-3, 7), ... , (4, 0)\).

Step 5: Draw the Graph

Once all points are plotted, draw a smooth curve through the points to represent the function \(h(x) = 16 - x^2\). This graph will form a parabola opening downwards with its vertex at \(x = 0\).

Key Concepts

Parabola

When it comes to graphing quadratic functions, understanding the nature of parabolas is key. A parabola is a U-shaped curve that can open upwards or downwards.
In the given exercise, the function is defined as \(h(x) = 16 - x^2\). This is a classic example of a downward-opening parabola because the coefficient of \(x^2\) is negative (-1).
When graphing a parabola, it's important to note its symmetry about the vertical line that passes through its vertex. This symmetry simplifies the graphing process.
Start by selecting a few values just before and after the vertex to get a sense for the curve. Parabolas also have a property called the axis of symmetry, which directly impacts their shape. The axis of symmetry is vertical and for a parabola given by \(y = ax^2 + bx + c\) takes the form \(x = -\frac{b}{2a}\). In our equation, however, since \(b=0\), the axis is simply \(x=0\).
This characteristic helps ensure that both sides of the parabola are mirror images of one another.

Table of Values

Creating a table of values is an important step in graphing parabolas. It serves as a guide to plotting points on a coordinate plane.
Start by choosing a range of \(x\)-values. Make sure to include the vertex and values on either side to accurately represent the parabola's shape.

• For the function \(h(x) = 16 - x^2\), it's a good idea to keep the \(x\)-values around the vertex easy to calculate, such as -4 to 4.


• Next, compute the \(y\)-values using the expression for the function. Replace each \(x\) with its respective value, and solve.


• The results will form pairs of points like \((-4, 0), (-3, 7), \ldots, (4, 0)\).

These points are crucial as they give a precise path to sketch the curve. Represent these on a graph by plotting them directly, helping to ensure accuracy before drawing the full curve.

Vertex of Parabola

The vertex is a pivotal point on the parabola where it turns, which represents either the maximum or minimum value of the function. For the equation \(h(x) = 16 - x^2\), the vertex is particularly special.

• Since the coefficient of \(x^2\) is negative, this parabola opens downwards, making the vertex its highest point, or maximum.


• The vertex can be directly identified by looking at the constants in the function. Here, the \(y\)-coordinate of the vertex, \(h(x)\), reaches its maximum at 16 when \(x = 0\).

This translates to the vertex being located at the point \((0, 16)\).
The vertex gives valuable information. It is not only the peak but also the axis of symmetry lies at this \(x\)-coordinate, making it easier to graph.
Understanding the vertex allows you to predict the general direction and shape of the parabola's arc on the graph. It’s integral to accurately sketching and interpreting quadratic functions.