Problem 22 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=9-x^{2} $$

Step-by-Step Solution

Answer

The x-intercepts are (3,0) and (-3,0); y-intercept is (0,9).

Step 1: Understanding the Function

The given equation is a quadratic function in the form of \(y = 9 - x^2\). This is a downward-opening parabola because the coefficient of \(x^2\) is negative.

Step 2: Creating a Table of Values

To find the values of y for various values of x, we create a table. Select a range of x-values that are symmetric around zero, such as \(-3, -2, -1, 0, 1, 2, 3\). Calculate corresponding y-values:- For \(x = -3\), \(y = 9 - (-3)^2 = 0\)- For \(x = -2\), \(y = 9 - (-2)^2 = 5\)- For \(x = -1\), \(y = 9 - (-1)^2 = 8\)- For \(x = 0\), \(y = 9 - 0^2 = 9\)- For \(x = 1\), \(y = 9 - 1^2 = 8\)- For \(x = 2\), \(y = 9 - 2^2 = 5\)- For \(x = 3\), \(y = 9 - 3^2 = 0\)

Step 3: Plotting Points and Sketching the Graph

Using the table of values, plot the points \((-3,0), (-2,5), (-1,8), (0,9), (1,8), (2,5), (3,0)\) on the coordinate plane. Sketch a symmetrical parabolic curve passing through these points.

Step 4: Finding the x-Intercepts

The x-intercepts are found where \(y = 0\). Set the equation \(9 - x^2 = 0\) and solve:- \(x^2 = 9\)- \(x = \pm 3\)Therefore, the x-intercepts are \((3, 0)\) and \((-3, 0)\).

Step 5: Finding the y-Intercept

The y-intercept is found where \(x = 0\). Substitute \(x = 0\) into the equation:- \(y = 9 - 0^2 = 9\)The y-intercept is \((0, 9)\).

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that we often encounter in quadratic functions. In the quadratic equation form, it is represented as \(y = ax^2 + bx + c\). The nature of the parabola, whether it opens upwards or downwards, is determined by the coefficient of the \(x^2\) term. If the coefficient is positive, the parabola opens upwards, shaped like the letter "U". Conversely, if the coefficient is negative, as in \(y = 9 - x^2\), it opens downwards, looking like an upside-down "U". This is why our parabola for the given function is downward-opening.

One of the key features of a parabola is its vertex, which is the highest or lowest point, depending on the direction it opens. For the function \(y = 9 - x^2\), the vertex is at the point \( (0, 9) \), making it the highest point on the parabola.

X-Intercepts

X-intercepts are the points where the graph of the function crosses the x-axis. At these points, \(y = 0\). To find the x-intercepts for the function \(y = 9 - x^2\), we set the equation to zero: \(9 - x^2 = 0\).

Solving this, we rearrange the equation to: \(x^2 = 9\). Taking the square root of both sides, we find two solutions: \(x = 3\) and \(x = -3\). Therefore, the x-intercepts are the points \( (3, 0) \) and \((-3, 0)\).

These points are important because they tell us where the parabola touches or crosses the x-axis. For quadratic functions, this can indicate the roots or solutions of the equation.

Y-Intercept

The y-intercept is where a graph crosses the y-axis. This occurs when \(x = 0\). To find the y-intercept of our function \(y = 9 - x^2\), we substitute \(x = 0\) into the equation.

Simplifying this, we get: \(y = 9 - 0^2 = 9\). Thus, the y-intercept is \( (0, 9) \).

The y-intercept gives us a specific point on the graph, serving as a helpful starting point when sketching. It's where the parabola reaches its maximum or minimum value, depending on if it opens downwards or upwards. In this case, since the parabola opens downwards, it's the maximum point.

Graph Sketching

Graph sketching involves plotting points and accurately representing the graph of a function on a coordinate plane. For quadratic functions like \(y = 9 - x^2\), we use key points such as the vertex, x-intercepts, and y-intercepts to guide the shape and position of the graph.

Here’s how you can approach sketching the given function:

• Create a table of values by selecting a range of x-values: \(-3, -2, -1, 0, 1, 2, 3\).
• For each x-value, compute the corresponding y-value.
• Plot these points on the coordinate plane: \((-3,0), (-2,5), (-1,8), (0,9), (1,8), (2,5), (3,0)\).

The symmetry of a parabola allows you to efficiently sketch the curve. Remember, the graph should be smooth, going through all plotted points in a U-shape. The vertex, x-intercepts, and y-intercept help anchor the graph, giving you a clear framework to work from.