Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\) if \(a=4\) : (a) \(f(x)=\log _{a} x\) (b) \(f(x)=-\log _{a} x\) (c) \(f(x)=2 \log _{a} x\) (d) \(f(x)=\log _{a}(x+2)\) (e) \(f(x)=\left(\log _{a} x\right)+2\) (f) \(f(x)=\log _{a}(x-2)\) (g) \(f(x)=\left(\log _{a} x\right)-2\) (h) \(f(x)=\log _{a}|x|\) (i) \(f(x)=\log _{a}(-x)\) (j) \(f(x)=\log _{a}(3-x)\) (k) \(f(x)=\left|\log _{a} x\right|\) (l) \(f(x)=\log _{1 / a} x\)

Step-by-Step Solution

Answer

Sketch each graph based on transformations described for \( a = 4 \).

Step 1: Basic Logarithmic Function

The basic function is given by \( f(x) = \log_{a} x \). When \( a = 4 \), this function passes through \( (1,0) \) since \( \log_{4}(1) = 0 \). It increases as \( x \) increases and flattens, approaching infinity.

Step 2: Reflected Logarithmic Function

For \( f(x) = -\log_{a} x \), the graph of \( \log_{a} x \) is reflected in the x-axis. This means that as \( x \) increases, \( f(x) \) decreases.

Step 3: Scaled Logarithmic Function

In the case of \( f(x) = 2\log_{a} x \), the graph of \( \log_{a} x \) is stretched vertically by a factor of 2. It will grow faster as \( x \) increases.

Step 4: Shifted Logarithmic Function Right

For \( f(x) = \log_{a}(x+2) \), the graph is shifted 2 units to the left. So, everything is moved towards the left by 2 units.

Step 5: Shifted Up Logarithmic Function

In \( f(x) = \log_{a} x + 2 \), the log graph is shifted upwards by 2 units. Thus, the whole graph moves up by 2 units.

Step 6: Shifted Logarithmic Function Left

The function \( f(x) = \log_{a}(x-2) \) shifts the basic graph 2 units to the right. This requires \( x > 2 \).

Step 7: Shifted Down Logarithmic Function

For \( f(x) = \log_{a} x - 2 \), the graph is the basic graph shifted downwards by 2 units. This affects the \( y \)-intercept to be \( -2 \) when \( x = 4 \).

Step 8: Absolute Logarithmic Function

In \( f(x) = \log_{a}|x| \), the function is divided into two pieces: for \( x > 0 \), it is \( \log_{a} x \), and for \( x < 0 \), it is the reflection of \( \log_{a}(-x) \) about the y-axis. Only defined for \( x eq 0 \).

Step 9: Negatively Scaled Logarithmic Function

For \( f(x) = \log_{a}(-x) \), the graph will be the reflection of the basic log graph across the y-axis, only for \( x < 0 \).

Step 10: Horizontal Logarithmic Shift

The expression \( f(x) = \log_{a}(3-x) \) shifts \( \log_{a} x \) horizontally. Convert to \( \log_{a}(-x+3) \), shifting everything left by 3 units.

Step 11: Absolute Magnitude Logarithmic Function

For \( f(x) = |\log_{a}(x)| \), take the basic graph and make the entire curve positive so it reflects in the x-axis for negative values of \( x \).

Step 12: Inverse Base Logarithmic Function

In \( f(x) = \log_{1/a} x \), the graph behaves like an inverse logarithm owing to \( \frac{1}{a} = \frac{1}{4} \), so the curve is essentially the mirror of the base \( a \). It will decrease as \( x \) increases.

Key Concepts

Graph Transformations

Graph transformations involve altering the basic graph of a function in several ways without changing its overall shape. Consider the basic logarithmic function \( f(x) = \log_{a} x \). Its graph starts from \((1, 0)\) and rises, flattening as \(x\) increases. When you're transforming this graph, you might be moving it up, down, left, or right, or even flipping and stretching it.

Some key transformations include:

• Vertical shifts: Moving the graph up or down.
• Horizontal shifts: Moving the graph left or right.
• Reflections: Flipping the graph across an axis.
• Scaling: Stretching or compressing the graph vertically or horizontally.

Understanding the impact of these transformations on logarithmic graphs can help you predict how a basic function will behave once altered.

Logarithmic Base Change

Changing the base of a logarithmic function drastically alters its graph. If you have \( f(x) = \log_{a} x \) and convert it to \( f(x) = \log_{1/a} x \), the graph's shape and direction change as well. Here's what happens when you change the base:

- A larger base \(a\) usually causes the graph to rise slower, and the graph of \(\log_{a} x\) increases as \(x\) increases.
- Inverting the base to \( 1/a \) results in a graph that decreases as \( x \) increases, since \( \log_{1/a} x \) is the reverse.

The critical point to remember is that changing the base affects how steep the graph looks and its directional growth.

Absolute Value Function

The absolute value function is about taking the positive value of the output, regardless of its sign. For example, with \( f(x) = |\log_{a} x| \), you take the normal logarithmic function and reflect any negative parts upward. This means:

• If \( \log_{a} x > 0 \), \( f(x) = \log_{a} x \).
• If \( \log_{a} x < 0 \), \( f(x) = -\log_{a} x \).

This transformation changes the graph by flipping any parts below the x-axis to above it, ensuring all values are non-negative. This can be a helpful way to visualize symmetric properties of graphs and make sure no values dip below the axis.

Function Reflection

Reflections are about flipping the graph over a particular axis. In logarithmic functions, consider \( f(x) = -\log_{a} x \), which reflects the graph over the x-axis. Let's see what happens here:

- The graph that was previously increasing now decreases because the y-values are inverted.
- This transformation effectively makes all positives negative and all negatives positive for \( f(x) \), altering the way the function behaves entirely.

Another type is reflecting around the y-axis, such as with \( \log_{a}(-x) \). Here, the graph shifts its location from positive x-values to negative ones. Recognizing reflections can provide insight into function behavior, especially in terms of symmetry relative to axes.

Vertical and Horizontal Shifts

Shifting functions either vertically or horizontally changes their position without altering their overall shape. When applying horizontal shifts in functions such as \( f(x) = \log_{a}(x + 2) \), the whole graph moves to the left because you're adding a value inside the logarithm:

- \( +2 \) shifts left, whereas \( -2 \) indicates a shift to the right.

Vertical shifts, as seen in \( f(x) = \log_{a} x + 2 \), raise the entire graph up by 2 units.

In these transformations:

• Horizontal shifts involve changing input values, moving the graph along the x-axis.
• Vertical shifts move the graph up or down the y-axis, altering output values.

Visualizing these shifts can significantly aid in understanding how equations translate into graphical language.