Problem 70 - College Algebra

Question

Describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$g(x)=5(x+3)^{2}-2$$

Step-by-Step Solution

Answer

The function is a vertically stretched, horizontally translated, and vertically shifted quadratic.

Step 1: Identify the Toolkit Function

The toolkit function for this transformation is the basic quadratic function, which is \( f(x) = x^2 \). Our goal is to understand how \( g(x) \) is a transformation of this function.

Step 2: Locate the Horizontal Translation

In the function \( g(x) = 5(x+3)^2 - 2 \), the expression \((x+3)\) indicates a horizontal translation. This means the graph of the basic function \( f(x) = x^2 \) has been shifted to the left by 3 units.

Step 3: Identify the Vertical Translation

The constant \(-2\) in the function \( g(x) = 5(x+3)^2 - 2 \) indicates a vertical translation. The whole graph is shifted downwards by 2 units.

Step 4: Recognize the Vertical Stretch

The coefficient 5 in front of the \((x+3)^2\) term suggests a vertical stretch. This means the graph is stretched away from the x-axis by a factor of 5, making it steeper.

Step 5: Combine the Transformations

Applying these transformations in sequence: first horizontally shift the graph of \( f(x) = x^2 \) left by 3 units, then stretch it vertically by a factor of 5, and finally shift it down by 2 units to complete the transformation to \( g(x) \).

Step 6: Sketch the Transformed Graph

Now, sketch the graph based on transformations:- Start with the basic parabolic graph of \( f(x) = x^2 \).- Shift the vertex from (0,0) to (-3,0).- Stretch it vertically, making it narrower (as if you are pulling it upwards).- Move the vertex 2 units down to (-3, -2). The new graph has its vertex at (-3, -2), is narrow due to the stretch, and opens upwards.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that can be written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The simplest form of a quadratic function is the toolkit function \( f(x) = x^2 \), which is a parabola opening upwards with its vertex at the origin, (0,0).

Quadratic functions have many unique features:

• The graph is a symmetrical curve called a parabola.
• The vertex is the highest or lowest point, depending on the direction of the parabola.
• The axis of symmetry runs vertically through the vertex, often defined as \( x = -\frac{b}{2a} \) for the general quadratic equation.
• The parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \).

Understanding the basic shape and features of the quadratic function is essential before applying any transformations.

Horizontal Translation

A horizontal translation alters the location of a graph along the x-axis. For a quadratic function like \( g(x) = (x + 3)^2 \), the expression \( (x + 3) \) indicates a horizontal shift. Specifically,

• If we see \( (x + 3) \), it implies moving the graph left by 3 units.
• Conversely, \( (x - 3) \) would mean moving right by 3 units.

To visualize this, consider the basic parabola \( f(x) = x^2 \):

• Its vertex starts at (0,0).
• A horizontal translation moves the vertex along the x-axis.

In our transformed function \( g(x) = 5(x+3)^2 - 2 \), the graph of the basic function shifts left by 3 units, placing the new vertex at (-3,0) before applying any other transformations.

Vertical Translation

Vertical translation refers to shifting the graph up or down along the y-axis. For the quadratic function \( g(x) = 5(x+3)^2 - 2 \), the term \(-2\) indicates a vertical translation.

• If the constant is negative, like \(-2\), the graph moves downward by that number of units.
• If the constant were positive, the graph would shift upward.

Visualizing this, take the parabolic graph post-horizontal shift \( (-3,0) \):

• Now, shift the entire graph down 2 units.
• This moves the vertex to the new point (-3, -2).

Vertical translations do not affect the shape or width of the graph but adjust its position up or down.

Vertical Stretch

A vertical stretch changes the steepness of a graph. It involves "stretching" the graph away from the x-axis, making it either taller or more compressed.For the function \( g(x) = 5(x+3)^2 - 2 \), the coefficient 5 signifies a vertical stretch.

• A value greater than 1 (like 5) implies the graph is stretched upwards, making it steeper.
• Conversely, a value less than 1 but greater than 0 (such as 0.5) would compress it, making the graph wider.

To understand this practically:

• The basic parabola \( f(x) = x^2 \) becomes steeper when multiplied by 5.
• This results in a "narrower" appearance because the sides of the parabola approach the y-axis more sharply.

After applying all transformations (horizontal shift, vertical stretch, and vertical translation), the graph of \( g(x) \) effectively changes shape and position, reflecting these effects.