Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-30, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\)-intercepts. $$ y=4 x^{2}-1 $$

Step-by-Step Solution

Answer

The graph is a parabola symmetric about the y-axis with y-intercept (0, -1) and x-intercepts (\(\pm\frac{1}{2}, 0\)).

Step 1: Identify Symmetries

To determine symmetry, check if the equation is symmetric about the y-axis, x-axis, or origin. For y-axis symmetry, replace x with -x and check if the equation remains unchanged. The equation becomes \(y=4(-x)^2-1=4x^2-1\), which is the same as the original. Thus, the graph is symmetric about the y-axis.

Step 2: Find the Y-Intercept

The y-intercept occurs where x = 0. Substitute x = 0 into the equation: \(y=4(0)^2-1=-1\). Therefore, the y-intercept is (0, -1).

Step 3: Find the X-Intercepts

The x-intercepts occur where y = 0. Set the equation to 0: \(0=4x^2-1\). Solve for x by adding 1 to both sides, yielding \(4x^2=1\), then divide by 4: \(x^2=\frac{1}{4}\). Taking the square root gives \(x=\pm\frac{1}{2}\). Thus, the x-intercepts are \((\frac{1}{2}, 0)\) and \((-\frac{1}{2}, 0)\).

Step 4: Plot the Graph

Using the symmetry about the y-axis, the y-intercept (0, -1), and the x-intercepts (\(\pm\frac{1}{2}, 0\)), sketch the parabola. The parabola opens upwards because the coefficient of \(x^2\) is positive. Plot these points and draw a symmetric curve through them.

Key Concepts

Symmetry in Functions

In mathematics, symmetry refers to a balanced and proportional similarity found in two halves of an object or equation. For functions, detecting symmetry is a key way to simplify graphing. The most common types of symmetry are around the y-axis, x-axis, or origin.
To determine if a function is symmetric about the y-axis, substitute \(x\) with \(-x\). For the provided equation \(y = 4x^2 - 1\), substituting \(x\) with \(-x\) gives the same equation: \(y = 4(-x)^2 - 1 = 4x^2 - 1\).
This consistency shows that the graph is symmetric about the y-axis. Such a symmetry can ease graph plotting significantly since you only need to calculate function values for half of the graph, knowing the other half will mirror it.

Finding Intercepts

Intercepts are key points where the curve crosses the axes. Every quadratic function will have an equation in the form of \(y = ax^2 + bx + c\). Calculating the intercepts helps you understand where the graph "interacts" with its surroundings.

• Y-Intercept: To find the y-intercept, set \(x = 0\) and solve for \(y\). For our equation, substituting \(x = 0\) gives \(y = 4(0)^2 - 1 = -1\). So, the y-intercept is the point \( (0, -1) \).
• X-Intercepts: To find where the graph crosses the x-axis, set \(y = 0\) and solve for \(x\). For our function, \(0 = 4x^2 - 1\) becomes \(x^2 = rac{1}{4}\). Solving gives \(x = rac{1}{2}\) and \(x = -rac{1}{2}\). Thus, x-intercepts are \( (rac{1}{2}, 0) \) and \( (-rac{1}{2}, 0) \).

These intercepts form the basis for sketching the function's graph, showing where the curve starts and ends along these axes.

Quadratic Equations

Quadratic equations are a central concept in algebra and take the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. These equations describe a parabola, a symmetrical curve that can open upwards or downwards depending on the sign of \(a\).

For the equation \(y = 4x^2 - 1\), the coefficient \(a = 4\) is positive, indicating that the parabola opens upwards. This particular quadratic has no linear \(b\) term, which simplifies plotting and finding vertex points.
Quadratic equations are used in various applications, from calculating trajectories to optimizing financial profits. Understanding their structure aids in manual graphing and in the algebraic manipulation necessary for them.