Problem 13 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ y=x^{2}-9 $$

Step-by-Step Solution

Answer

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

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(y = x^2 - 9\). This simplifies to \(y = 0^2 - 9\). Therefore, \(y = -9\). Thus, the y-intercept is the point \((0, -9)\).

Step 2: Find the x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation \(y = x^2 - 9\). We have \(0 = x^2 - 9\). Add 9 to both sides to get \(x^2 = 9\). Taking the square root of both sides, we get \(x = 3\) or \(x = -3\). Thus, the x-intercepts are the points \((3, 0)\) and \((-3, 0)\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree. They have the general form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable.
In our specific example, the quadratic equation is \(y = x^2 - 9\). Here, this simplifies to the form \(ax^2 + bx + c\) where \(a = 1\), \(b = 0\), and \(c = -9\). This tells us that the graph of this quadratic is a parabola opening upwards, as the coefficient \(a\) is positive.
One critical aspect of quadratic equations is finding their solutions, which are often the points where they intersect the x-axis. These solutions are also known as the x-intercepts or "roots" of the equation.
When solving a quadratic, set the equation equal to zero and solve for \(x\). The solutions can be found through factoring, completing the square, or using the quadratic formula.

Graphing Functions

Graphing functions involves plotting the equation on a set of axes. This provides a visual representation of mathematical relationships, which can be very helpful for understanding how functions behave.
For the quadratic equation \(y = x^2 - 9\), the graph is a parabola. To make the graph, start by determining key points, such as the x- and y-intercepts. The y-intercept is the value where the graph crosses the y-axis. This occurs when \(x = 0\). In our equation, substituting \(x = 0\) gives \(y = -9\), leading to the y-intercept at the point \((0, -9)\).

• To find x-intercepts, set \(y = 0\) and solve for \(x\). Originally, \(0 = x^2 - 9\), which simplifies to \(x = \pm 3\). This gives two x-intercepts at points \((3, 0)\) and \((-3, 0)\).
• Using these intercepts, you can start to plot the function on the graph. The points serve as a guide to sketch the parabola's curve.

Parabolas

A parabola is a symmetric, U-shaped curve which is the graph of a quadratic function. Every parabola has key features that help us understand its shape and position on the graph.
In the standard form \(y = ax^2 + bx + c\), if \(a > 0\), the parabola opens upwards like a "smile", and if \(a < 0\), it opens downwards like a "frown". For our equation \(y = x^2 - 9\), the parabola opens upwards because \(a = 1\).
Additional characteristics of parabolas include the vertex, which is either the highest or lowest point on the graph, depending on the direction the parabola opens. Here, the vertex is at \((0, -9)\) since the equation is in the form \(x^2 - k\), and the value \(k = -9\) gives the lowest point.

• Parabolas are also symmetric about their vertical axis, known as the axis of symmetry. In this equation, the axis of symmetry is the vertical line \(x = 0\).
• Identifying intercepts, vertex, and symmetry helps in sketching an accurate graph of the parabola.