Problem 12 - 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. \(f(x)=x^{2}-x-2 ;\) right 2 units, upward 3 units

Step-by-Step Solution

Answer

The shifted equation is \(g(x) = (x - 2)^2 - (x - 2) + 1\). Graph both \(f(x) = x^2 - x - 2\) and \(g(x)\) on the same plane.

Step 1: Understand the Graph Transformations

To shift the graph of a function, we use transformations. Shifting a graph to the right involves replacing \(x\) with \(x - a\), and shifting it upward involves adding a constant \(b\). In our case, shifting right by 2 units means replacing \(x\) with \(x - 2\), and shifting upward by 3 units means adding 3 to the entire function.

Step 2: Apply Horizontal Shift Transformation

For the horizontal shift right by 2 units, replace \(x\) with \(x - 2\) in the function \(f(x) = x^2 - x - 2\). So the transformed function becomes: \(f(x - 2) = (x - 2)^2 - (x - 2) - 2\).

Step 3: Apply Vertical Shift Transformation

Add 3 to the transformed function \(f(x - 2)\) to shift it upward by 3 units. This gives: \(g(x) = (x - 2)^2 - (x - 2) - 2 + 3\).

Step 4: Express the Shifted Function's Equation

The equation of the shifted graph is \(g(x) = (x - 2)^2 - (x - 2) + 1\). This function represents the graph of \(f(x)\) shifted 2 units to the right and 3 units up.

Key Concepts

Horizontal Shift

When we talk about a horizontal shift in graph transformations, we are adjusting the graph horizontally along the x-axis. Imagine you have a quadratic function like the one given:

• The general form of the function is \( f(x) = x^2 - x - 2 \).
• A horizontal shift to the right by 2 units involves replacing every occurrence of \( x \) in the equation with \( (x - 2) \).

This means that the x-values are increased by 2. As a result, the entire graph moves to the right side of the coordinate plane without altering its shape. If instead we wanted to shift the graph to the left, we would use \( (x + a) \), where \( a \) is the number of units we want to shift it to the left. Consider the impact of increasing or decreasing the x-value in real-world problems, like shifting the time point forward or backward.

Vertical Shift

Vertical shifts move the graph up or down along the y-axis. This happens when you add or subtract a constant from the entire function. In our quadratic function example:

• Once you have adjusted the function horizontally to \( (x - 2)^2 - (x - 2) - 2 \), you add 3 to the entire function to perform a vertical shift.

The transformation carries the graph upwards by three units. This adjustment causes each point on the graph to move up, eventually creating a new graph that is parallel to the original, but higher along the y-axis. If you wished to shift the graph downwards, you would subtract a constant instead. Vertical shifts are intuitive as they mimic real-life scenarios, like increasing a baseline measurement or adjusting profits upward by a fixed amount.

Quadratic Function

Quadratic functions are functions of the form \( f(x) = ax^2 + bx + c \). These functions typically produce parabolic graphs, which are either u-shaped or n-shaped, depending on the sign of the leading coefficient, \( a \). In our case, the function is \( f(x) = x^2 - x - 2 \):

• This function is a standard quadratic function because its highest degree term is \( x^2 \).
• A positive coefficient in front of \( x^2 \) indicates the parabola opens upwards.

When dealing with shifts in quadratic functions, it's important to understand how these transformations interact with the inherent features of parabolas, like their vertex and axis of symmetry. Quadratic functions have a predictable pattern and they appear frequently in physics and economics, representing processes like projectile motion or cost minimization.