Problem 52 - College Algebra

Question

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$h(x)=2+\log _{2} x$$

Step-by-Step Solution

Answer

The x-intercept of the function \(h(x)=2+\log _{2} x\) is \(x=0.25\) and the vertical asymptote is \(x=0\).

Step 1: Graphing the Original Function

Begin by graphing the original function \(f(x)=\log _{2} x\). A logarithm function \(log_a(x)\) typically has a vertical asymptote at \(x=0\) and an x-intercept at \(x=1\). This is because the line \(x=0\) is where the function is undefined and the line \(x=1\) is where the function equals 0.

Step 2: Applying Transformations

The given function \(h(x)=2+\log _{2} x\) has a transformation compared to the original function. The '2' added to the function translates the graph of the function upward by two units. This means all the points on the original graph \(f(x)=\log _{2} x\) are shifted up by two units to obtain the graph of \(h(x)=2+\log _{2} x\).

Step 3: Determining the x-intercept

The x-intercept of a function is the value of \(x\) for which the function equals 0. So we want to find \(x\) for which \(h(x)=2+\log _{2} x=0\). Solve this equation for \(x\). Subtract 2 from both sides to get \(\log _{2} x= -2\). In exponent form, this equation is saying \(2^(-2) = x\). So, \(x= 0.25\) is the x-intercept.

Step 4: Determining the Vertical Asymptote

The vertical asymptote of the logarithm function did not change due to the transformation. It will remain at \(x=0\).

Key Concepts

Graph Transformations

When dealing with logarithmic functions, graph transformations can significantly alter the appearance of the graph without changing its fundamental properties. For the base function, such as \(f(x)=\log_{2} x\), the graph has a vertical asymptote at \(x=0\) and an x-intercept at \(x=1\). This is a classic shape of a logarithmic curve. However, when we transform the function by adding a number, like \(h(x)=2+\log_{2} x\), we are essentially shifting the graph.

• Adding 2 to the entire function results in a vertical shift upwards by 2 units. This affects each and every point on the graph, moving the entire curve parallel to its original position.
• This transformation does not alter the x-intercept’s x-coordinate, but the y-value changes due to the upward shift.

Understanding how transformations work is key. They allow you to predict how the graph will look without plotting numerous points.

Vertical Asymptote

In logarithmic functions, a vertical asymptote represents a line that the graph approaches but never touches or crosses. It is a critical feature of logarithms. For the basic form \(f(x)=\log_{2} x\), the vertical asymptote appears at \(x=0\). This is because the logarithm of zero or a negative number is undefined.With transformations, noticing changes in the vertical asymptote is important:- Adding a constant to the function, as in \(h(x)=2+\log_{2} x\), does not affect the vertical asymptote.- The graph still approaches the line \(x=0\), meaning the vertical asymptote stays at this location.Always keep this in mind while working with logarithmic transformations. When the transformation involves shifts only in the vertical direction, the asymptote stays unchanged.

X-Intercept

The x-intercept is the point where the graph of a function crosses the x-axis. In the context of logarithmic functions like \(f(x)=\log_{2} x\), the x-intercept is usually \(x=1\) because this is the point where the logarithm of 1 equals zero.When finding the x-intercept for a transformed function like \(h(x)=2+\log_{2} x\), it requires solving for when the function equals zero:
1. Set the function \(h(x)=0\), transforming the equation to \(2+\log_{2} x=0\).2. Subtract 2 from both sides to arrive at \(\log_{2} x=-2\).3. Translate this to the exponential form: \(x=2^{-2}=0.25\).This means the x-intercept is \(x=0.25\), different from the untransformed graph. Finding the x-intercept involves understanding both the effect of transformations and the process of solving logarithmic equations.