Problem 14 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. The graph of \(g(x)=4+\log x\) is similar to the graph of \(f(x)=\log x,\) but it is translated 4 units ____. b. The graph of \(g(x)=\log _{4}(x+2)\) is similar to the graph of \(f(x)=\log _{4} x,\) but it is translated 2 units to the ____.

Step-by-Step Solution

Answer

a. Upward. b. Left.

Step 1: Translate the Logarithm Equation

The function given is \(g(x) = 4 + \log x\) and it should be compared to \(f(x) = \log x\). Imagine \(f(x)\) and consider what adding a constant outside of a log function does. Adding a number outside the logarithmic term results in a vertical translation of the graph of the function.

Step 2: Determine the Vertical Shift Impact

Since the function \(g(x) = 4 + \log x\) has a '+4' added to the original \(\log x\), the graph of \(f(x) = \log x\) is translated vertically upwards by 4 units.

Step 3: Translate the Logarithm Equation with Added Constant

The function \(g(x) = \log_4(x + 2)\) should be compared to \(f(x) = \log_4 x\). When you add a number inside, with the variable in the logarithm function, it results in a horizontal shift in the opposite direction of the sign.

Step 4: Determine the Horizontal Shift Impact

Since \(g(x) = \log_4(x + 2)\) includes \(+2\) inside with the variable, it causes a leftward shift by 2 units because adding inside the argument of a logarithm translates the graph in the negative direction on the x-axis.

Key Concepts

Vertical Translation

A vertical translation involves shifting a graph up or down without affecting its shape. Think of it as moving the entire graph vertically along the y-axis. In our exercise, the function is given as \(g(x) = 4 + \log x\). This is quite similar to the basic logarithmic function \(f(x) = \log x\), but with a crucial difference: the addition of a constant outside the logarithm. Adding \(4\) to the entire function means that every point on the original graph of \(\log x\) moves up by 4 units. So, if you started with a point \((x, \log x)\), it shifts to \((x, \log x + 4)\).

• This type of shift does not affect the x-values; only the y-values change.
• Thus, every point on the graph moves uniformly upward.
• The new graph remains a logarithmic function, but at a higher y-level than before.

Horizontal Translation

Horizontal translation involves moving a graph left or right on the x-axis. This kind of shift is mainly controlled by terms added or subtracted from the x-variable inside the function expression. For the function \(g(x) = \log_4(x + 2)\), the original function \(f(x) = \log_4 x\) undergoes a shift due to the \(+2\) inside the argument. Here's where it can be confusing: you'd naturally think adding \(+2\) would move things to the right, but in graph translations, it’s the opposite. Adding \(+2\) to the \(x\)-variable causes the entire graph to shift left by that amount.

• Every point moves left, stacked precisely 2 units farther than before.
• This shift doesn't alter the height or the general shape of the function.
• An easy way to remember this is: opposite inside, same outside for direction of shifts.

Graph Transformations

Graph transformations refer to manipulating the position or shape of the graph. Transformations can be categorized as either transformations affecting position or those affecting the shape. In our specific exercise, we see examples of transformations affecting the graph's position: vertical and horizontal translations. Both translation types alter where the graph sits on the coordinate system without changing its shape. This means if you were drawing the graph by hand, you could sketch the basic function first and simply slide it to its new position.

• Vertical translations move the entire graph up or down.
• Horizontal translations push it left or right.
• These adjustments are key in fine-tuning graphs to display data correctly or better fit particular scenarios.

Understanding these transformation rules helps in graphing functions accurately and interpreting the effects of changes in equations.