Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation, if possible. $$y=4-x^{2}$$

Step-by-Step Solution

Answer

The x-intercepts are 2 and -2. The y-intercept is 4.

Step 1: Find the x-intercepts

To find the x-intercept(s), set \(y = 0\) in the equation and solve for \(x\). So we get \(0 = 4 - x^2\), which simplifies to \(x^2 = 4\). Taking square roots of both sides we get \(x = 2\) or \(x = -2\). Hence, the x-intercepts are 2 and -2.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). This gives \(y = 4 - 0^2 = 4\). Hence, the y-intercept is 4.

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. To find the x-intercept, we set the value of y to zero. In the equation form, this means solving for x when y equals zero. In this example, with the equation \( y = 4 - x^2 \), to find the x-intercept, you replace y with zero, obtaining the setup \( 0 = 4 - x^2 \). Solving this, you first move \( x^2 \) to the other side of the equation, resulting in \( x^2 = 4 \). Then, you find the square root of both sides, yielding two potential solutions for x: \( x = 2 \) or \( x = -2 \). These solutions, \( (2,0) \) and \( (-2,0) \), represent the points where the graph hits the x-axis.

Understanding the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. To determine the y-intercept, set \( x = 0 \) in the equation and solve for y. For the equation \( y = 4 - x^2 \), this means substituting zero for x: \( y = 4 - 0^2 \), which simplifies to \( y = 4 \). Thus, the y-intercept is located at the point \( (0, 4) \). This point tells us where the graph begins on the y-axis. It's an important feature because it provides a starting point for graphing the equation on a coordinate plane.

The Quadratic Equation Demystified

A quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \). It is characterized by the highest power of the variable being two. The given equation \( y = 4 - x^2 \) is a quadratic equation because it can be written as \( -x^2 + 0x + 4 = 0 \). Quadratic equations often form a parabolic graph, which can either open upwards or downwards depending on the sign of the \( x^2 \) term. In this case, because the \( x^2 \) term is negative, the parabola opens downwards. This shape is a fundamental aspect of any quadratic function and determines the behavior and trajectory of its graph.

Graphing the Quadratic Equation

Graphing a quadratic equation involves plotting points derived from the equation on a coordinate plane and then drawing a smooth curve through these points to form a parabola. Start by identifying the intercepts which help anchor your graph. In our case, we have the x-intercepts at \( (2, 0) \) and \( (-2, 0) \), and the y-intercept at \( (0, 4) \). These points help in visualizing the basic structure of the parabola. When graphing, it's also beneficial to check additional points such as \( (1, 3) \) or \( (-1, 3) \), which are calculated using the original equation. Plotting these points will give a more precise shape of the graph. Remember, the curve is symmetrical with respect to the y-axis since \( y = 4 - x^2 \) is an even function, and ensure the graph appropriately displays this symmetry.