Question

Find the intercepts of the equation $y=x^{2}-9 .

Short answer

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

Step-by-step solution

Understand the Equation

The given equation is a quadratic equation in the standard form, where the equation is written as \(y = x^{2} - 9\). We need to find both the x-intercepts and the y-intercept.

Find the Y-Intercept

To find the y-intercept, we set \(x = 0\) in the equation. Substitute \(x = 0\) into the equation: \(y = 0^{2} - 9 = -9\). Therefore, the y-intercept is at the point \((0, -9)\).

Find the X-Intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\). \(0 = x^{2} - 9\). Add 9 to both sides: \(x^{2} = 9\). Take the square root of both sides: \(x = \text{±}3\). So, the x-intercepts are at the points \((3, 0)\) and \((-3, 0)\).

Key concepts

x-intercepts

In mathematics, the x-intercepts of a graph are the points where the graph crosses the x-axis. For the given quadratic equation, the x-intercepts can be found by setting the value of y equal to zero and solving for x. This is done because at the x-intercepts, the value of y is zero.

Let's break that down:

• We start with the equation given: \(y = x^{2} - 9\).
• To find the x-intercepts, we set \(y = 0\) and solve for x: \(0 = x^{2} - 9\).
• Next, we add 9 to both sides to isolate the \(x^{2}\) term: \(x^{2} = 9\).
• The final step is to take the square root of both sides, giving us two solutions: \(x = 3\) and \(x = -3\).

Therefore, the x-intercepts are at the points (3, 0) and (-3, 0). Understanding how to find the x-intercepts helps us better visualize and understand the shape and position of the quadratic graph.

y-intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. This point can be found by setting the value of x equal to zero and solving for y. This is done because at the y-intercept, the value of x is zero.

Here's the step-by-step process:

• We start with the equation given: \(y = x^{2} - 9\).
• To find the y-intercept, we set \(x = 0\) and solve for y: \(y = 0^{2} - 9\).
• This simplifies to \(y = -9\).

Therefore, the y-intercept is at the point (0, -9). Knowing the y-intercept is important because it gives us a fixed point through which the graph passes, helping to sketch the general shape of the quadratic function.

solving quadratic equations

Solving quadratic equations involves finding the values of x that make the equation true. Quadratics are polynomial equations of degree two, and they generally appear in the form \(ax^2 + bx + c = 0\). There are several methods to solve quadratic equations, but for this specific exercise, we used factoring.

The given quadratic equation is already simplified to \(x^2 - 9 = 0\). We solve it as follows:

• First, isolate the \(x^2\) term by moving all other terms to the other side: \(x^2 - 9 = 0\) becomes \(x^2 = 9\).
• Next, take the square root of both sides. This gives us two solutions because the square root of a positive number has both a positive and a negative solution: \(x = \text{±} \text{3}\).

Therefore, the solutions to the quadratic equation \(x^{2} - 9 = 0\) are \(x = 3\) and \(x = -3\).

Understanding how to solve quadratic equations is fundamental in algebra because it allows us to find the points of intersection with the x-axis, which, in turn, helps us understand the behavior of the graph.