Problem 23 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$g(x)=-3(x+2)^{2}-4$$

Step-by-Step Solution

Answer

The underlying basic function is \(f(x) = x^{2}\). The transformations applied to the basic function for obtaining \(g(x)=-3(x+2)^{2}-4\) are a vertical reflection, a vertical stretch by a factor of 3, a 2-unit shift to the left, and a 4-unit shift downwards.

Step 1: Identify the Basic Function

The first task is to identify the basic function that is being transformed. In this case, recognized that the function \(g(x)=-3(x+2)^{2}-4\) is a transformed version of the basic quadratic function \(f(x) = x^{2}\).

Step 2: Identify the Transformations

We can see that our function has a negative sign in front of the \(x^{2}\) term, which means there is a vertical reflection, a multiplier of 3 to the \(x^{2}\) term, indicating a vertical stretch by a factor of 3, the term \(+2\) inside the brackets, denoting a horizontal shift to the left by 2 units, and the term \(-4\) indicating a vertical shift downwards by 4 units.

Step 3: Apply the Transformations to the Basic Function

To sketch the graph of this function, start with the graph of the basic function \(x^{2}\) and apply the transformations. First, reflect the graph over the x-axis due to the negative sign. Then, stretch the graph vertically by a factor of 3. Now, shift the graph to the left by 2 units, and finally, shift it down by 4 units. The resulting graph will represent the function \(g(x)=-3(x+2)^{2}-4\).

Key Concepts

Basic Quadratic Function

Quadratic functions are among the most essential relationships in algebra. They express relationships where a variable is squared. The basic form of a quadratic function is given by \( f(x) = x^2 \). This simple equation yields a parabolic curve that opens upwards. The graph is symmetric around the y-axis and its vertex, the lowest point for this standard equation, lies at the origin (0, 0).

When you're working with quadratic equations, the key feature to look for is the exponent of 2 on the variable, signaling you are dealing with a parabola. Altering this base form by adding, subtracting, or multiplying different numbers leads to transformations that shift, stretch, compress, or reflect the parabola across the graph.

Vertical Reflection

A vertical reflection in function transformations involves flipping the graph of the function over the x-axis. This happens when a negative sign is introduced in front of the quadratic term \( x^2 \).

In our function \( g(x) = -3(x+2)^2 - 4 \), there is a negative sign in front of the quadratic term, indicating a vertical reflection. Instead of opening upwards like the parent function \( f(x) = x^2 \), the graph opens downwards.

• The graph maintains the same shape but is inverted around the x-axis.
• This causes the vertex, which is typically the minimum point in \( f(x)=x^2 \), to become the maximum point in the reflected graph.

Horizontal Shift

A horizontal shift occurs when a constant is added or subtracted inside the squared term of a quadratic equation. This transformation moves the parabola left or right across the graph.

For our function \( g(x) = -3(x+2)^2 - 4 \), the \(+2\) indicates a shift. The shift will be to the left, as the general rule for horizontal shifts is that \( f(x+c) \) shifts the graph \( c \) units to the left.

• This means that each point of the graph of the basic parabola \( x^2 \) will move 2 units to the left.
• The vertex position changes accordingly from (0, 0) to (-2, something based on other transformations).

Vertical Stretch

Vertical stretching involves changing the steepness or "width" of the parabola by multiplying \( x^2 \) by a constant factor. This makes the parabola appear narrower.

In \( g(x) = -3(x+2)^2 - 4 \), the coefficient \(-3\) represents a significant vertical stretch. Instead of the typical rise and run of \( x^2 \), each point on the graph is pushed further from the x-axis by a factor of 3.

• If the multiplier is greater than 1, the function is vertically stretched, meaning it gets narrower.
• This transformation affects only the vertical dimension, not the symmetry of the parabola or the horizontal positions of its points.