Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the \(x\) -intercept \((\mathrm{s})\) and \(y\) -intercepts(s) of the graph of \(4 x^{2}+9 y=72\)

Short answer

x-intercepts: \( \pm 3\sqrt{2}\), y-intercept: 8

Step-by-step solution

Identify the equation

The given equation is \(4x^2 + 9y = 72\).

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). So, substitute \(y = 0\) into the equation: \[4x^2 + 9(0) = 72\]. This simplifies to \[4x^2 = 72\]. Divide both sides by 4 to get \[x^2 = 18\]. Take the square root of both sides: \[x = \pm \sqrt{18}\]. Thus, the x-intercepts are \(x = \sqrt{18}\) and \(x = -\sqrt{18}\), which simplifies to \(x = \pm 3\sqrt{2}\).

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation: \[4(0)^2 + 9y = 72\]. This simplifies to \[9y = 72\]. Divide both sides by 9 to get \[y = 8\]. Thus, the y-intercept is \(y = 8\).

Key concepts

finding x-intercept

In order to find the x-intercepts of a graph, you need to set the y-value to zero. This is because x-intercepts are points where the graph crosses the x-axis, and at these points, the y-value is always zero. To find the x-intercept for the equation given, follow these steps:

• Start with the equation: \(4x^2 + 9y = 72\).
• Set \(y = 0\): \(4x^2 + 9(0) = 72\).
• Now, solve for \(x\): \(4x^2 = 72\).

Divide both sides by 4 to isolate \(x^2\): \(x^2 = 18\). The final step is to take the square root of both sides: \(x = \pm \sqrt{18}\). Simplifying gives us: \(x = \pm 3 \sqrt{2}\). This means that the graph crosses the x-axis at \(x = 3 \sqrt{2}\) and \(x = -3 \sqrt{2}\).

finding y-intercept

Finding the y-intercept is a similar process, but instead of setting \(y\) to zero, you set \(x\) to zero. This is because y-intercepts are points where the graph crosses the y-axis, and at these points, the x-value is always zero.
Follow these steps to find the y-intercept for the given equation:

• Start with the equation: \(4x^2 + 9y = 72\).
• Set \(x = 0\): \(4(0)^2 + 9y = 72\).
• Simplify to: \(9y = 72\).
• Divide both sides by 9 to isolate \(y\).

This gives you: \(y = 8\). Hence, the graph crosses the y-axis at \(y = 8\).

solving quadratic equations

Quadratic equations are polynomials of the form \(ax^2 + bx + c = 0\). Solving these equations often gives us the x-intercepts of the graph. Let's break down some common methods to solve quadratic equations:

• **Factoring**: Rewriting the equation as a product of two binomials, if possible.
• **Completing the Square**: Transforming the equation into a perfect square trinomial.
• **Quadratic Formula**: Using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions.

For example, for the equation \(4x^2 = 72\), we solved it by isolating \(x^2\) and then taking the square root.
This is a simple application, but other methods might be needed for more complex quadratics.
Understanding these methods is essential for finding x-intercepts and solving many algebraic problems.