Problem 15 - 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. \(f(x)=3 x^{2}+2 x-5\) (a) Shifted left 3 units (b) Shifted downward 4 units

Step-by-Step Solution

Answer

\(g(x) = 3x^2 + 20x + 24\)

Step 1: Understand Horizontal Shifts

A horizontal shift of a function involves adding or subtracting a constant to the input variable. If you need to shift the function to the left by 3 units, you substitute \(x\) with \(x + 3\). This leads to the transformation of the function from \(f(x)\) to \(f(x+3)\).

Step 2: Apply Horizontal Shift to f(x)

To shift \(f(x)=3x^2+2x-5\) left by 3 units, replace every \(x\) in the function with \(x+3\):\[f(x+3) = 3(x+3)^2 + 2(x+3) - 5\]

Step 3: Calculate the Horizontal Shift Transformation

Now, expand the expression to determine the new function:Start by expanding \((x+3)^2\):\[(x+3)^2 = x^2 + 6x + 9\]Thus, the function becomes:\[3(x^2 + 6x + 9) + 2(x+3) - 5\]Expand and combine terms:\[= 3x^2 + 18x + 27 + 2x + 6 - 5\]\[= 3x^2 + 20x + 28\]

Step 4: Understand Vertical Shifts

A vertical shift involves adding or subtracting a constant from the entire function. To shift downward by 4 units, subtract 4 from the whole expression for \(f(x)\).

Step 5: Apply Vertical Shift to Transformed Function

From the function we derived in Step 3 (\(g(x) = 3x^2 + 20x + 28\)), subtract 4 to shift it downward:\[g(x) = 3x^2 + 20x + 28 - 4\]\[= 3x^2 + 20x + 24\]

Step 6: Write the Final Function

The function \(g(x)\), which represents \(f(x)\) shifted left by 3 units and downward by 4 units, is:\[g(x) = 3x^2 + 20x + 24\] This is the final expression that includes both the horizontal and vertical shifts applied.

Key Concepts

Horizontal Shift

Understanding a horizontal shift is key when transforming a function. Imagine you want to move a graph left or right. This is done by changing the input part of the function.
To shift a graph horizontally:

• Add a value inside the function's variable to move it left.
• Subtract a value to move it right.

For example, if we have a function such as a quadratic function like
\(f(x) = 3x^2 + 2x - 5\), a left shift by 3 units involves replacing every instance of \(x\) with \(x+3\).
This makes sense because, effectively, you're "starting" the function 3 units earlier, so it moves left on the graph.
This results in the transformation:
\(f(x+3) = 3(x+3)^2 + 2(x+3) - 5\).
Expanding this transformation gives the new function:
\(3x^2 + 18x + 27 + 2x + 6 - 5\), which simplifies to
\(3x^2 + 20x + 28\).
This new equation describes the same shape, simply shifted 3 units left.

Vertical Shift

A vertical shift occurs when you change the height of a graph. You do this by adding or subtracting a number from the entire function's output.
This doesn't change the shape, just moves it up or down in the grid.
To shift a function vertically:

• Add to move it up.
• Subtract to move it down.

In our example, we take the horizontal shift result \(g(x) = 3x^2 + 20x + 28\) and make it move 4 units down.
This means we have to subtract 4 from everything, resulting in:
\(g(x) = 3x^2 + 20x + 28 - 4\).
After performing the subtraction, the new equation becomes:
\(g(x) = 3x^2 + 20x + 24\).
This operation keeps the structure of the parabola the same, just adjusted its position on the vertical axis.

Quadratic Function

Quadratic functions are the ones where the highest power of the variable is squared. They are often written in the form of:
\(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The shape is a parabola, which is a smooth, symmetrical curve. Depending on the sign of \(a\), it can open upwards (\(a > 0\)) or downwards (\(a < 0\)).
Features of quadratic functions include:

• The vertex: the highest or lowest point of the parabola.
• The axis of symmetry: a vertical line that passes through the vertex.
• The y-intercept: where the function crosses the y-axis.

In transformations like horizontal and vertical shifts, these features move;
however, the parabola’s shape remains unchanged. This makes understanding shifts essential because it tells us how graphs overlap or separate in space, maintaining their core characteristics.