Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=-\frac{1}{2} x^{3} $$

Step-by-Step Solution

Answer

The graph is a downward-opening cubic curve with intercepts at (0,0).

Step 1: Identify the Equation

The given equation is \( y = -\frac{1}{2} x^3 \). This is a cubic equation which opens downward due to the negative coefficient.

Step 2: Find the Y-Intercept

The y-intercept occurs when \( x = 0 \). Plug in \( x = 0 \) into the equation: \( y = -\frac{1}{2} (0)^3 = 0 \). Thus, the y-intercept is \((0, 0)\).

Step 3: Find the X-Intercepts

The x-intercepts occur when \( y = 0 \). Set the equation to zero: \( 0 = -\frac{1}{2} x^3 \). Solving for \( x \), we find \( x^3 = 0 \), which gives \( x = 0 \). Thus, the x-intercept is \( (0, 0) \).

Step 4: Plot Simple Points

To sketch the graph, let's find some additional points. For example, when \( x = 1 \), \( y = -\frac{1}{2} (1)^3 = -\frac{1}{2} \). When \( x = -1 \), \( y = -\frac{1}{2} (-1)^3 = \frac{1}{2} \). Plot these points: \( (1, -\frac{1}{2}) \) and \( (-1, \frac{1}{2}) \).

Step 5: Draw the Graph

Plot the intercepts \((0, 0)\) and the additional points \((1, -\frac{1}{2})\) and \((-1, \frac{1}{2})\) on a coordinate plane. Draw a smooth curve through these points reflecting the nature of a cubic graph, and ensure it passes through the origin with the characteristic cubic shape opening downward.

Step 6: Label the Graph

Clearly label the x-axis, y-axis, and the intercepts on your graph. Make sure to mark the point \((0,0)\) as both the x- and y-intercept.

Key Concepts

Graphing Equations

Graphing equations involves creating a visual representation of a mathematical equation on a coordinate plane. This process helps to understand the behavior of the equation by providing a graphical perspective. For cubic equations like \(y = -\frac{1}{2} x^3\), the graph typically exhibits a characteristic smooth curve that opens either upwards or downwards depending on the sign of the leading coefficient. Here, the negative sign indicates the graph opens downward.
To graph such equations, always start by identifying key points like the x- and y-intercepts, as they help to anchor the shape and position of the curve. Furthermore, plotting additional points by substituting different values of \(x\) helps in drawing a more accurate curve. The curve should reflect the nature of cubic functions—often resembling an S-shape—yet in this case, opening downward due to the negative sign.

X-Intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of \(y\) is zero. For the cubic equation \(y = -\frac{1}{2} x^3\), setting \(y = 0\) gives:

• \(0 = -\frac{1}{2} x^3\)
• Solve for \(x\): \(x^3 = 0\)
• Therefore, \(x = 0\)

Thus, the graph crosses the x-axis at \((0, 0)\). This indicates that the origin is both an x-intercept and a root of the equation. Understanding x-intercepts is crucial in graphing, as they give insight into the solutions or zeros of the equation.

Y-Intercepts

The y-intercept is where a graph crosses the y-axis, meaning the value of \(x\) is zero at this point. For the equation \(y = -\frac{1}{2} x^3\), finding the y-intercept involves substituting \(x = 0\) into the equation:

• \(y = -\frac{1}{2} (0)^3 = 0\)

This means the y-intercept is also at \((0, 0)\). The y-intercept provides another anchor point that helps to guide the drawing of the curve on the coordinate plane. It's important to label this intercept clearly on the graph to illustrate where the curve passes through the y-axis.

Coordinate Plane

The coordinate plane is a two-dimensional surface where graphs of equations are drawn. It consists of a horizontal line called the x-axis and a vertical line called the y-axis. Where these axes intersect is known as the origin, marked as \((0, 0)\). The coordinate plane allows you to plot points using ordered pairs \((x, y)\) that indicate positions.
When graphing the equation \(y = -\frac{1}{2} x^3\), it's vital to properly label the axes and plot points accurately. Start with the intercepts you've calculated and then add additional points, such as \((1, -\frac{1}{2})\) and \((-1, \frac{1}{2})\), to provide a clear trajectory of the cubic curve. Through plotting on the coordinate plane, the relationship described by the equation comes to life, helping to visualize its dynamics and behavior.