Problem 77 - Calculus Early Transcendentals: Pearson New International Edition

Question

Using the same axes, draw the conics \(y=\) \(\pm\left(a x^{2}+1\right)^{1 / 2}\) for \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\) using \(a=\) \(-2,-1,-0.5,-0.1,0,0.1,0.6,1 .\) Make a conjecture about how the shape of the figure depends on \(a\).

Step-by-Step Solution

Answer

The shape changes based on 'a', from inverted (negative 'a') to wider (positive 'a') conics.

Step 1: Understand the Equation

The equation given is for two conics: \( y = \pm (ax^2 + 1)^{1/2} \). This represents a hyperbola as it contains the term \(x^2\). It will open upwards and downwards, as indicated by the positive and negative square root terms.

Step 2: Analyze the Influence of 'a'

Examine how different values of \(a\) affect the shape. If \(a = 0\), the equation simplifies to \(y = \pm 1\), a straight line. For negative \(a\), the parabola-like shape is inverted, and for positive \(a\), it becomes wider. Increasing \(a\) or decreasing \(a\) in magnitude affects the 'width' of the curve.

Step 3: Graph for Each 'a' Value

For each given \(a\): \(-2, -1, -0.5, -0.1, 0, 0.1, 0.6, 1\), substitute into the equation and plot \(y = \pm (ax^2 + 1)^{1/2}\) on the axes within the domain \(-2 \leq x \leq 2\) and range \(-2 \leq y \leq 2\). The curves will appear progressively closer to or further from the \(y\)-axis depending on the value of \(a\).

Step 4: Interpret the Graphs

For negative \(a\), the curves are sharp and more close (inverted hyperbolas), as \(a\) approaches zero, the hyperbolas widen becoming flatter at \(a=0\). Positive \(a\) values will stretch the hyperbola more, making it wider.

Step 5: Make a Conjecture

Based on the graphics, we can conjecture that for negative \(a\), the curves are more densely packed and appear inverted, and as \(a\) becomes positive, the curves widen. Thus, the value of \(a\) controls the opening and width of each branch of the conics.

Key Concepts

Hyperbolas

Hyperbolas are one of the four types of conic sections, which also include circles, ellipses, and parabolas. A hyperbola can be visualized as two symmetric, open curves extending away from each other. In the equation form given here,

• The hyperbola is vertical, meaning it opens upwards and downwards.
• The expressions \( y = \pm (ax^2 + 1)^{1/2} \) define two mirror-image parts of the hyperbola, due to the \( \pm \) sign.
• The squared term \( x^2 \) signifies that the hyperbola branches grow infinitely, as x stretches toward infinity.

Hyperbolas have unique properties such as asymptotes, lines, the graph approaches but never touches. These help guide the overall shape and position of the hyperbola. The specific term involving a square root in this exercise simplifies the visualization of how each piece of the hyperbola behaves.

Effect of Parameters

Parameters in an equation significantly shape the curve of the graph it represents. Here, the parameter is 'a' found in the hyperbola equation

• When \( a = 0 \), the hyperbola simplifies to the lines \( y = \pm 1 \). This happens because the \( ax^2 \) term disappears, leaving a flat line at \( y = 1 \) and \( y = -1 \).
• For negative values of 'a',
  • The hyperbola becomes steeper and inverted, causing the curves to approach each other more closely. This shows a contraction across the x-axis.
• Conversely, when 'a' is positive, the magnitude increase causes the hyperbola to stretch and widen, particularly opening more towards the y-axis.

Understanding parameter effects gives you control over graph manipulation, revealing the mathematical structure within diverse equations.

Graphing Equations

Graphing, in this context, requires plotting the equation \( y = \pm (ax^2 + 1)^{1/2} \) for different 'a' values. Here are the essential steps:

• Select the range: in this exercise, it is specified as \( -2 \leq x \leq 2 \) and \( -2 \leq y \leq 2 \).
• Substitute each value of 'a' into the equation and calculate corresponding y-values for x-values within the given range.
• Plot the points for each resulting equation. \( y \) will take both positive and negative values, representing the two branches of the hyperbola.

Each curve represents how the hyperbola develops based on particular 'a' values. Pay attention to how they move, twist, or transform. Continue practicing with diverse ranges and parameters to fully appreciate how mathematical graphing serves as a bridge between equations and their visual interpretations.