Problem 21 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

X-intercepts: 3 and -3; Y-intercept: -9.

Step 1: Create a Table of Values

Choose a range of x-values to substitute into the equation to get corresponding y-values. A symmetrical selection around zero often works well for quadratic functions. Use values like \(-3, -2, -1, 0, 1, 2, 3\). Calculate the corresponding y-value for each x-value using the equation \(y = x^2 - 9\).

Step 2: Calculate Y-Values

Substitute each selected x-value into \(y = x^2 - 9\):- For \(x = -3\), \(y = (-3)^2 - 9 = 0\).- For \(x = -2\), \(y = (-2)^2 - 9 = -5\).- For \(x = -1\), \(y = (-1)^2 - 9 = -8\).- For \(x = 0\), \(y = 0^2 - 9 = -9\).- For \(x = 1\), \(y = 1^2 - 9 = -8\).- For \(x = 2\), \(y = 2^2 - 9 = -5\).- For \(x = 3\), \(y = 3^2 - 9 = 0\).

Step 3: Sketch the Graph

Plot the points \((-3, 0), (-2, -5), (-1, -8), (0, -9), (1, -8), (2, -5), (3, 0)\) on a coordinate plane. Draw a smooth curve through these points. The symmetry of the points about the y-axis should form a parabola opening upwards.

Step 4: Find the X-Intercepts

To find the x-intercepts, set \(y = 0\) and solve \(x^2 - 9 = 0\). Factoring gives \((x - 3)(x + 3) = 0\), so the x-intercepts are \(x = 3\) and \(x = -3\).

Step 5: Find the Y-Intercept

The y-intercept is found by setting \(x = 0\) in the equation. This gives \(y = 0^2 - 9 = -9\). Therefore, the y-intercept is \(y = -9\).

Key Concepts

X-Intercepts

To determine the x-intercepts of a quadratic function, set the value of \(y\) equal to 0 in the given equation. For instance, with the equation \(y = x^2 - 9\), setting \(y = 0\) yields \(x^2 - 9 = 0\). This step will help you identify the points where the parabola crosses the x-axis.

Next, solve the equation \(x^2 - 9 = 0\) by factoring. Rewrite it as \((x - 3)(x + 3) = 0\). From this factorization, you find two solutions: \(x - 3 = 0\) and \(x + 3 = 0\). Solving these, you get \(x = 3\) and \(x = -3\).

Thus, the x-intercepts are located at the points \((3, 0)\) and \((-3, 0)\). These are the points where the graph intersects the x-axis. Remember that finding x-intercepts is crucial for understanding the graph's layout on the coordinate plane.

Y-Intercepts

The y-intercept of a function is where the graph crosses the y-axis. To find it, set \(x = 0\) in the quadratic equation. For the equation \(y = x^2 - 9\), substitute \(x = 0\) to find \(y\).

This calculation becomes straightforward: \(y = 0^2 - 9 = -9\). As a result, the y-intercept is found at the point \((0, -9)\).

Finding the y-intercept is an easy yet essential step when sketching the graph of a quadratic function. It provides a fixed point that is usually a base for graphing the shape of the parabola.

Parabola

A parabola is a symmetric curve that is the graphical representation of a quadratic function. Its standard form is typically expressed as \(y = ax^2 + bx + c\). The equation \(y = x^2 - 9\) is a simple form with \(a = 1\), \(b = 0\), and \(c = -9\).

Parabolas display interesting features. They have a vertex, which is the highest or lowest point on the graph. For \(y = x^2 - 9\), the vertex is at \( (0, -9) \).

Additionally, parabolas are symmetric about a vertical line, known as the axis of symmetry. In this case, the axis of symmetry is the y-axis, due to the absence of the linear \(b\) term in the equation.

Understanding the shape and properties of a parabola is crucial in graphing quadratic functions, as it helps visualize how the graph behaves.

Graphing Techniques

Graphing a quadratic function involves certain techniques that make the process simple and intuitive. One effective method is creating a table of values. Choose values of \(x\) around the vertex to find corresponding \(y\) values. For the equation \(y = x^2 - 9\), selecting \(x\) values like \(-3, -2, -1, 0, 1, 2, 3\) gives symmetry and clarity.

Plot these points on a graph:

• \((-3, 0)\)
• \((-2, -5)\)
• \((-1, -8)\)
• \((0, -9)\)
• \((1, -8)\)
• \((2, -5)\)
• \((3, 0)\)

Once plotted, draw a smooth curve connecting the dots. The resulting U-shaped graph visually represents a parabola opening upwards.

Relying on symmetry and precise plotting aids in accurately sketching the graph. This technique underscores the importance of understanding the relationship between values in quadratic equations.