Problem 18 - College Algebra with Modeling and Visualization

Question

Write a formula for a function \(g\) whose graph is similar to \(f(x)\) but satisfies the given conditions. Do not simplify the formula. \(f(x)=5 x^{2}-3\) (a) Shifted left 10 units and downward 6 units (b) Shifted right 1 unit and upward 10 units

Step-by-Step Solution

Answer

(a) \(g(x) = 5(x + 10)^2 - 9\), (b) \(g(x) = 5(x - 1)^2 + 7\).

Step 1: Understanding the Problem

We need to modify the graph of the quadratic function \(f(x) = 5x^2 - 3\) according to specific transformations. These transformations involve horizontal and vertical shifts. Our task is to find the new expressions for the function based on these shifts.

Step 2: Transformation Rules

When shifting a function horizontally, substitute \(x\) with \(x - a\) to shift right \(a\) units, or with \(x + a\) to shift left \(a\) units. To shift a function vertically, add or subtract a constant \(b\) to shift up or down respectively.

Step 3: Shift Left and Down

For part (a), we shift \(f(x)\) left 10 units and downward 6 units. The horizontal shift replaces \(x\) with \(x + 10\), and the vertical shift subtracts 6 from the entire function. Therefore, \(g(x) = 5(x + 10)^2 - 3 - 6\).

Step 4: Shift Right and Up

For part (b), we shift \(f(x)\) right 1 unit and upward 10 units. Here, the horizontal shift replaces \(x\) with \(x - 1\), and the vertical shift adds 10 to the function. Thus, \(g(x) = 5(x - 1)^2 - 3 + 10\).

Key Concepts

Horizontal Shifts

Horizontal shifts in functions occur when you move the graph left or right on the coordinate plane. For a quadratic function like \( f(x) = 5x^2 - 3 \), this involves changing the expression inside the parenthesis. When you replace \( x \) with \( x + a \), you're shifting the function to the left by \( a \) units. This might seem counterintuitive because you're adding \( a \), but in the context of function transformations, it's a leftward movement. Conversely, replacing \( x \) with \( x - a \) shifts the graph to the right by \( a \) units. - An example is if we shift \( f(x) \) to the left 10 units, the function becomes \( 5(x + 10)^2 - 3 \). - To shift right by 1 unit, the transformation would be \( 5(x - 1)^2 - 3 \). These horizontal modifications are powerful for altering the location of the function along the x-axis while keeping its shape intact.

Vertical Shifts

Vertical shifts change the position of the function along the y-axis. These are simpler compared to horizontal shifts because they merely involve addition or subtraction outside the function itself. When you add a constant \( b \) to the function \( f(x) \), it moves the graph upwards by \( b \) units, whereas subtracting \( b \) moves it downwards. This occurs because you are directly raising or lowering the output value of the function.For example, with our function \( f(x) = 5x^2 - 3 \):

• Shifting downward by 6 units, as in part (a), means subtracting 6 from the whole function, resulting in: \( 5(x+10)^2 - 3 - 6 \).
• Conversely, shifting upward by 10 units, as in part (b), means adding 10, changing the function to: \( 5(x-1)^2 - 3 + 10 \).

Through vertical shifts, you can easily move the graph up or down without affecting its structure.

Function Transformations

Combining the methods of horizontal and vertical shifts is a common way to manipulate function graphs effectively. These transformations allow you to reposition a quadratic function wherever needed on the coordinate plane.A transformation often involves both types of shifts: adjusting the \( x \) variable for horizontal movement and altering the overall expression for vertical movement. Both actions combine into the function's new formula, which consists of the modified parent function and additional constants. - In part (a) of the problem, you executed both a leftward and a downward shift, yielding \( g(x) = 5(x+10)^2 - 9 \).- For part (b), the rightward and upward shifts generated \( g(x) = 5(x-1)^2 + 7 \).By using these transformations, you can customize any quadratic function's placement without changing its form. This method maintains the original function's shape, merely relocating it within the plane.