Problem 10 - College Algebra with Modeling and Visualization

Question

Find an equation that shifts the graph of \(f\) by the desired amounts. Do not simplify. Graph \(f\) and the shifted graph in the same \(xy\)-plane. \(f(x)=3 x-4 ;\) left 3 units, upward 1 unit

Step-by-Step Solution

Answer

The shifted function equation is \( 3(x+3) - 3 \).

Step 1: Understanding Horizontal Shifts

To shift the graph of a function horizontally, we modify the function's variable in the function expression. Shifting left by 3 units means we replace \( x \) with \( x + 3 \) in the function. So, starting with \( f(x) = 3x - 4 \), the horizontally shifted function becomes \( 3(x + 3) - 4 \).

Step 2: Understanding Vertical Shifts

To shift the graph of a function vertically, we add or subtract a constant to the output of the function. Since we need to move the graph upward by 1 unit, we add 1 to the function. For our current function \( 3(x + 3) - 4 \), this becomes \( 3(x + 3) - 4 + 1 \).

Step 3: Write the Final Shifted Function

Combine the changes from both horizontal and vertical shifts. The equation for the shifted graph is \( f(x + 3) + 1 = 3(x + 3) - 4 + 1 \). You do not need to simplify this expression as per the instructions.

Step 4: Final Function Equation

The equation for the shifted function is \( 3(x + 3) - 4 + 1 \). Therefore, \( 3(x + 3) - 3 \) represents the final shifted equation.

Key Concepts

Horizontal Shifts

When it comes to transforming graphs, understanding horizontal shifts is key. These shifts change the position of a graph along the x-axis without affecting its shape. To achieve a horizontal shift, we adjust the variable within the function itself. For example, when we shift the graph to the left by 3 units, we replace each occurrence of the variable \(x\) with \(x + 3\). This might seem counterintuitive since adding usually feels like moving to the right. However, in this transformation, the function's response is to nudge the graph left.

• For a shift left: Replace \(x\) with \(x + a\).
• For a shift right: Replace \(x\) with \(x - a\).

Thus, for the function \(f(x) = 3x - 4\), a leftward shift results in \(f(x) = 3(x+3) - 4\). It is crucial to recognize that these shifts do not distort or stretch the graph but merely transport it horizontally along the x-axis.

Vertical Shifts

Vertical shifts adjust the graph's position up or down along the y-axis. This type of shift is relatively straightforward as it involves simply adding or subtracting a constant from the function's output. If we want to move the graph upwards by a specified number of units, we add to the function. Conversely, moving it down requires subtraction.

• For an upward shift: Add \(c\) to the function.
• For a downward shift: Subtract \(c\) from the function.

In our example, after executing a horizontal shift, we further alter the function \(3(x + 3) - 4\) by adding 1 to account for one unit upward. This modification results in the expression \(3(x + 3) - 4 + 1\). The transformation signifies increasing all y-values by a constant, maintaining the graph's slope and linear properties intact.

Linear Functions

Linear functions are equations representing straight lines when plotted on a graph. They have a general form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These elements determine the line's steepness and starting point on the y-axis.
In the context of graph transformations, especially with linear functions like \(f(x) = 3x - 4\), any horizontal or vertical shift will maintain the straight line's integrity.

• Slope (\(m\)): Controls how steep the line is. Here, a slope of 3 means the graph rises 3 units for every additional unit along the x-axis.
• Y-intercept (\(b\)): This is the line's initial crossing point on the y-axis. In this case, it starts at \(-4\).

Applying shifts doesn't alter the line's slope. Instead, it simply repositions the entire line horizontally or vertically on the graph. Thus, though shifts change where the line lies, they won't convert its fundamental linear character.