Problem 17 - 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. \(f(x)=2 x^{2}-4 x+1\) (a) Shifted right 2 units and upward 4 units. (b) Shifted left 8 units and downward 5 units.

Step-by-Step Solution

Answer

For part (a), \(g(x) = 2(x-2)^2 - 4(x-2) + 5\); for part (b), \(g(x) = 2(x+8)^2 - 4(x+8) - 4\).

Step 1: Understanding Transformations

The function transformations involve shifting the graph horizontally and vertically. A shift right by 2 units involves subtracting 2 from the input, resulting in adjusting the function to become \(f(x-2)\). Similarly, a shift upward by 4 units involves adding 4 to the entire function.

Step 2: Apply Right Shift and Upward Shift

Apply the transformations for part (a) by substituting \(x\) with \(x-2\) and then adding 4 to the function:\[g(x) = f(x-2) + 4 = 2(x-2)^2 - 4(x-2) + 1 + 4\]. This gives us the new function form.

Step 3: Understanding Different Transformations

For part (b), shifting the function left by 8 units involves adding 8 to the input, resulting in \(f(x+8)\). Shifting downward by 5 units involves subtracting 5 from the entire function.

Step 4: Apply Left Shift and Downward Shift

Apply the transformations for part (b) by substituting \(x\) with \(x+8\) and subtracting 5 from the function: \[g(x) = f(x+8) - 5 = 2(x+8)^2 - 4(x+8) + 1 - 5\]. This is the new function for the given transformation.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function. They have the general form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The graph of a quadratic function is a parabola. It can open upwards or downwards depending on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

The vertex of the parabola gives us the maximum or minimum point. The axis of symmetry of a quadratic function is the line \(x = -\frac{b}{2a}\). This line divides the parabola into two mirror-image halves.
Quadratic functions can also exhibit transformations. These transformations change the position and shape of the parabola on the graph.
Understanding these basics will help in manipulating quadratic equations to get desired transformations.

Horizontal Shifts

Horizontal shifts move the graph of a function left or right along the x-axis. With quadratic functions, this involves changing the input \(x\) to \(x - h\). This adjustment shifts the graph:

• Right when \(h\) is positive.
• Left when \(h\) is negative.

For example, given \(f(x) = 2x^2 - 4x + 1\):

• Shifting right by 2 units, we replace \(x\) with \(x - 2\) to get \(f(x-2)\).
• Shifting left by 8 units, replace \(x\) with \(x + 8\) to get \(f(x+8)\).

These shifts do not alter the shape of the graph, only its position. It's crucial for students to see how the x-value changes impact the graph's orientation.

Vertical Shifts

Vertical shifts move the graph of a function up or down along the y-axis. This involves adding or subtracting a constant \(k\) from the whole function.

• Add \(k\) to shift the graph up.
• Subtract \(k\) to shift the graph down.

For the quadratic function \(f(x) = 2x^2 - 4x + 1\):

• Shifting the graph 4 units upward involves adding 4, so we have the transformation \(f(x) + 4\).
• Shifting 5 units downward involves subtracting 5, resulting in \(f(x) - 5\).

These vertical shifts modify the y-values, moving the graph up or down without changing its horizontal position or the parabola's shape.
Combining horizontal and vertical shifts can help position any function's graph exactly where needed on the coordinate grid.