Problem 58 - Calculus

Question

If \(g(x)=\ln (a x+2),\) where \(a \neq 0,\) what is the effect of increasing \(a\) on (a) The \(y\) -intercept? (b) The \(x\) -intercept?

Step-by-Step Solution

Answer

Increasing \( a \) does not change the y-intercept but moves the x-intercept closer to zero.

Step 1: Understanding the y-intercept

The y-intercept is the value of the function when \( x = 0 \). Thus, for \( g(x) = \ln(ax + 2) \), the y-intercept is computed as \( g(0) = \ln(2) \). The value of the intercept depends solely on the constant 2 and is independent of \( a \). Therefore, increasing \( a \) has no effect on the y-intercept.

Step 2: Finding the x-intercept

The x-intercept occurs when \( g(x) = 0 \). Setting \( g(x) = 0 \) gives us \( \ln(ax + 2) = 0 \), which implies \( ax + 2 = 1 \). Solving for \( x \) gives \( x = \frac{1-2}{a} = \frac{-1}{a} \).

Step 3: Analyzing the effect of increasing \( a \) on x-intercept

The calculated x-intercept is \( x = \frac{-1}{a} \). If \( a \) increases, the absolute value of \( \frac{-1}{a} \) decreases, meaning the x-intercept moves closer to zero. Thus, the x-intercept approaches the y-axis as \( a \) increases.

Key Concepts

Understanding the Y-Intercept

When working with logarithmic functions like \(g(x) = \ln(ax + 2)\), understanding the y-intercept is a key concept. The y-intercept represents where the function graph crosses the y-axis. You find the y-intercept by setting \(x = 0\) and solving for \(g(x)\).
In this case, substituting \(x = 0\) into the function gives us \(g(0) = \ln(2)\). Notice that the y-intercept is determined by the constant term 2. Hence, the y-intercept is \(\ln(2)\), irrespective of the parameter \(a\).
An important takeaway is that, since \(a\) is not part of the expression \(\ln(2)\), changing \(a\) does not affect the y-intercept at all. Whether \(a\) increases or decreases, the y-intercept remains constant at \(\ln(2)\). This illustrates that some elements in a logarithmic function can remain unchanged when parameters like \(a\) vary.

Finding the X-Intercept

The x-intercept in a function is another crucial point. It occurs where the graph crosses the x-axis, which means where \(g(x) = 0\). For the function \(g(x) = \ln(ax + 2)\), we set the equation for zero: \(\ln(ax + 2) = 0\).
To solve for \(x\), remember that \(\ln(y) = 0\) when \(y = 1\). That tells us \(ax + 2 = 1\). Solving for \(x\), we rearrange the equation: \[ax + 2 = 1 \ax = 1 - 2 \x = \frac{-1}{a}\]So, \(x = \frac{-1}{a}\) is the x-intercept. Here lies a vital detail: unlike the y-intercept, the x-intercept does depend on the parameter \(a\). This is because \(a\) is inside the expression being logged.
Incrementing \(a\) means that \(\frac{-1}{a}\) gets smaller in magnitude, which means the x-intercept moves closer to the y-axis—that is, towards zero.

Effect of Parameter Change

A comprehendible topic in function transformations is the effect of parameter changes on the graph behavior. Specifically for \(g(x) = \ln(ax + 2)\), we want to look at what happens when \(a\) changes.

• **Y-Intercept:** As previously stated, changing \(a\) has no effect on the y-intercept, which remains constant at \(\ln(2)\).
• **X-Intercept:** The x-intercept, on the other hand, is directly affected by \(a\). Increasing \(a\) decreases \(\left|\frac{-1}{a}\right|\), shifting the intercept toward zero.
• **General Graph Shape:** As \(a\) increases, the slope of \(ax+2\) before applying the logarithmic function grows steeper. This affects the spread of the log graph along the x-axis, making it appear more compressed.

Having a solid understanding of parameter impacts helps in graphical and analytical predictions of function behaviors. This also aids in gaining intuition on the dependency of function characteristics on specific parameters.