Problem 70 - College Algebra

Question

For the following exercises, 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 transformation involves a left shift by 3, a vertical stretch by 5, and a downward shift by 2.

Step 1: Identify the Toolkit Function

The given function is \( g(x) = 5(x+3)^2 - 2 \). The toolkit function here is \( f(x) = x^2 \), which is a basic quadratic function and represents a parabola.

Step 2: Identify Transformations

The function \( g(x) = 5(x+3)^{2} - 2 \) involves several transformations applied to the toolkit function \( f(x) = x^2 \):1. **Horizontal Shift**: \( (x+3)^2 \) represents a shift to the left by 3 units.2. **Vertical Stretch**: The coefficient 5 outside the squared term indicates a vertical stretch by a factor of 5.3. **Vertical Shift**: The \(-2\) at the end moves the graph down by 2 units.

Step 3: Sketch the Graph

To sketch the graph of \( g(x) = 5(x+3)^{2} - 2 \):1. Start with the basic parabola \( y = x^2 \).2. Shift the graph 3 units left to account for \( (x+3)^2 \).3. Apply the vertical stretch by multiplying the \( y \)-values by 5.4. Finally, shift the entire graph 2 units down.The vertex of the transformed parabola is at \((-3, -2)\). The graph opens upwards and is narrower than the basic parabola due to the vertical stretch.

Key Concepts

Quadratic Function

A quadratic function is one of the simplest and most fundamental types of polynomial functions. It is generally represented in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The graph of a quadratic function is a curve known as a parabola.
A parabola can open upwards or downwards:

• Upwards if \( a > 0 \)
• Downwards if \( a < 0 \)

The basic form or "toolkit" function for quadratics is \( f(x) = x^2 \), which creates a parabola with its vertex at the origin (0,0) and that opens upwards. This basic parabola is symmetrical about the y-axis, making it appear as a smooth, mirror-like curve.
For any quadratic, the vertex is a crucial point since it is the maximum or minimum point of the parabola. Finding transformations from this simple form allows us to understand different parabolas' shapes and positions in the coordinate plane.

Horizontal and Vertical Shifts

Transformations can shift the graph of a function horizontally and vertically. These transformations move the graph without altering its shape. In the given function \( g(x) = 5(x+3)^2 - 2 \), two shifts occur:

• Horizontal Shift: The term \( (x+3) \) indicates a shift to the left by 3 units. This might seem counterintuitive since it involves adding 3, but in the context of transformations, \( x + c \) generally results in shifting to the left if \( c > 0 \).
• Vertical Shift: The \,\( -2 \), at the end of the function, moves every point of the graph down by 2 units. This moves the entire graph without any distortion, altering only its vertical placement on the plane.

By applying these shifts, the vertex of the parabola moves from (0, 0) to (-3, -2). Both types of shifts help in positioning the parabola where it needs to be on the graph based on the context of the problem or application scenario.

Parabola Transformations

The transformations of parabolas involve changing their size, shape, and orientation on the graph. For the transformation described in \( g(x) = 5(x+3)^2 - 2 \), there are distinct changes from the toolkit quadratic function:

• Vertical Stretch: The coefficient 5 in front of \( (x+3)^2 \) stretches the parabola vertically. This means the parabola becomes narrower compared to \( y = x^2 \). The larger the value of the coefficient, the narrower the parabola, as each \( y \)-value is multiplied by this factor.
• Direction: The parabola in this example opens upwards due to a positive coefficient, consistent with the basic function \( x^2 \).

The combination of these transformations determines the parabola's final position and appearance on the graph. Visualizing these changes helps in sketching the function accurately and understanding how algebraic modifications influence geometric representations. This understanding can be vital in applications ranging from physics to engineering, where precise graph modeling is required.