Question

Use shifts and scalings to graph then given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed. $$g(x)=2(x+3)^{2}$$

Short answer

Answer: The transformations applied to the original function are a horizontal shift of 3 units to the left and a vertical scaling by a factor of 2. The horizontal shift is represented by the term (x+3) in the given function, while the vertical scaling is represented by the factor 2. These transformations shift the graph of the parabola 3 units to the left and stretch it vertically by a factor of 2.

Step-by-step solution

Identify the original function and the transformations

The given function is $$g(x)=2(x+3)^{2}$$ The original function is the quadratic function $$f(x)=x^2$$. The transformations applied to the original function are: 1. Horizontal shift: $$x+3$$ shifts the graph 3 units to the left. 2. Vertical scaling: Multiplying the function by 2 stretches the graph.

Apply the horizontal shift

To apply the horizontal shift, replace x with (x+3) in the original function: $$f(x+3)=(x+3)^2$$ This transformation shifts the graph of the original function 3 units to the left.

Apply the vertical scaling

Now apply the vertical scaling to the shifted function by multiplying it with 2: $$g(x)=2(x+3)^2$$ This transformation stretches the graph of the shifted function vertically by a factor of 2.

Plot the original and transformed functions

Now that we have applied both transformations, we can plot the original function $$f(x)=x^2$$ and the transformed function $$g(x)=2(x+3)^2$$. 1. Original function: plot the parabola $$f(x)=x^2$$. 2. Transformed function: plot the parabola $$g(x)=2(x+3)^2$$ after shifting it 3 units to the left and stretching it by a factor of 2 vertically.

Verify the graph with a graphing utility

Use a graphing utility to plot both the original function $$f(x)=x^2$$ and the transformed function $$g(x)=2(x+3)^2$$ to confirm that the transformations were applied correctly. The graph of $$g(x)$$ should be the graph of $$f(x)$$ shifted 3 units to the left and scaled vertically by a factor of 2.

Key concepts

Horizontal Shift

One of the transformations applied to a function is the horizontal shift. This affects the position of the graph along the x-axis. In the given exercise, for the function \( g(x) = 2(x+3)^2 \), we see \((x+3)\).
The \(+3\) indicates a movement to the left. This might seem counterintuitive because there's a plus sign, but it actually moves the graph in the opposite direction along the x-axis.

To understand this, think about solving a simple equation like \( (x + 3) = 0 \). Here, \( x = -3 \). This helps you see why the graph moves three units to the left. This type of shifting is essential when modifying or analyzing functions.

The rule of thumb for horizontal shifts is:

• \(x + n\) means shift left by \(n\) units.
• \(x - n\) means shift right by \(n\) units.

Observing how components inside the square or absolute value affect the graph is a powerful tool in graph transformations.

Vertical Scaling

Vertical scaling transforms the graph of a function by stretching or compressing it along the y-axis. In this exercise, the transformation happens when we multiply the quadratic expression \((x+3)^2\) by 2 to get \( g(x) = 2(x+3)^2 \).
By multiplying by 2, every y-value of the original function \( f(x) = x^2 \) is doubled.

To visualize vertical scaling, imagine stretching a rubber band vertically. This transformation affects the steepness or width of the parabola, making it narrower if the factor is greater than 1, as in this case.

• If the multiplier is greater than 1, the graph stretches (becomes steeper).
• If the multiplier is between 0 and 1, the graph compresses (becomes flatter).

Vertical scaling does not affect the x-values, only how high or low the points on the graph are plotted. This makes it crucial to consider when predicting graph shapes.

Quadratic Function

A quadratic function is defined as a polynomial of degree 2, often written in the form \( f(x) = ax^2 + bx + c \).
In its simplest form, like \( f(x) = x^2 \), the graph is a U-shaped curve called a parabola.

Quadratic functions are fundamental in mathematics because they describe a variety of real-world phenomena, including projectile motion and area calculations. The vertex of a parabola plays a key role, acting as either a maximum or minimum point of the function depending on the orientation of the parabola.

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

In the exercise, the original function \( f(x) = x^2 \) undergoes transformations to produce a new function. These transformations illustrate how modifications to the basic quadratic can result in a wide range of curves useful across different contexts. Understanding the form and shape of these functions is pivotal as they appear frequently in both theoretical and applied mathematics.