Question

In Exercises \(57-74\), sketch the graph of the equation. Look for extrema, intercepts, symmetry, and asymptotes as necessary. Use a graphing utility to verify your result. $$ y=4\left(1-\frac{1}{x^{2}}\right) $$

Short answer

The graph of the function \(y = 4 \left(1 - \frac{1}{x^{2}}\right)\) has no extrema or intercepts, is symmetric, decreases as we move away from the y-axis, and has a horizontal asymptote at \(y = 4\) and a vertical asymptote at \(x = 0\). The graph is a hyperbola opening upwards and downwards with the x-axis as its transverse axis.

Step-by-step solution

Identify the Range and Domain

The domain of the function is \(x ≠ 0\) because \(x = 0\) would make the denominator equal to zero, which is undefined in mathematics. The range of the function is \(y < 4\), \(y > 0\) because \(1 - \frac{1}{x^{2}}\) is always less than 1 but greater than 0 and it is multiplied by 4.

Look for Extrema

The function does not have any extrema (minima or maxima) because it is continually decreasing as we move away from the y-axis in either direction.

Check for Symmetry

The function is symmetric with respect to the y-axis because if we replace \(x\) with \(-x\), the function remains unchanged.

Find the Intercepts

The y-intercept does not exist because when \(x = 0\), \(y\) is undefined. There are no x-intercepts too because \(y = 4\left(1 - \frac{1}{x^{2}}\right) = 0\) yields no value of \(x\).

Look for Asymptotes

When \(x\) approaches 0, the value of \(y\) goes to infinity (positive or negative depending on direction of approach); thus \(x=0\) is a vertical asymptote. When \(x\) approaches infinity in positive or negative direction, the value of \(y\) approaches 4; thus \(y=4\) is a horizontal asymptote.

Sketch the Graph

Finally, after recognizing the characteristics of the function, a sketch of the graph can be made. The graph shows a curve decreasing from left to right, approaching the horizontal asymptote \(y=4\). We verify this by using a graphing utility.