Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-(x+3)^{2}-1\)

Step-by-Step Solution

Answer

The graph of \(y=-(x+3)^{2}-1\) is a downward-opening parabola that is shifted 3 units to the left and 1 unit down compared to the graph of \(y=x^{2}\).

Step 1: Identify the shape of the base function

The base function in this exercise is \(y = x^{2}\). This is a parabola that opens upward, with the origin (0,0) as its vertex.

Step 2: Identify the transformations

The function \(y=-(x+3)^{2}-1\) has three transformations applied to the base function: 1. The sign of x is changed, indicating a reflection across the y-axis.2. The \(x+3\) term represents a horizontal shift of the graph 3 units to the left.3. The -1 outside the square function represents a vertical shift down by 1 unit.4. The negative sign outside the square function indicates a reflection across the x-axis that flips the parabola so it opens downward.

Step 3: Sketch the graph

Applying the above transformations to the base function \(y=x^{2}\) gives the graph of \(y=-(x+3)^{2}-1\). Start with the base function, then apply the transformations one by one. First, shift the graph to the left by 3 units, then move it down by 1 unit, and finally reflect it across the x-axis.

Step 4: Compare with the original graph

The graph of \(y=-(x+3)^{2}-1\) is a downward-opening parabola that has been moved 3 units to the left and 1 unit down compared to the original graph of \(y=x^{2}\), which is an upward-opening parabola centered at the origin.

Key Concepts

Parabola Transformations

Parabola transformations involve modifying a standard parabola, which is typically represented as the function \(y = x^2\). These transformations allow us to change key features of the parabola, such as its direction, position, and shape. Transformations typically affect parabolas in the following ways:

• Direction of Opening: This refers to whether the parabola opens upwards or downwards. A positive coefficient in front of \(x^2\) opens upwards, while a negative coefficient opens downwards.
• Position: Position shifts can move the parabola left, right, up, or down, altering its vertex from the origin (0,0) to another location in the coordinate plane.
• Reflection: A reflection can occur across the x-axis or y-axis, effectively flipping the graph.

Understanding these transformations is crucial as they form the basis for graphing complex quadratic functions.

Reflection Across Axes

Reflecting a parabola across an axis changes its orientation. Let's break this down:

• Reflection across the x-axis: This flips the parabola upside down. For example, changing from \(y = x^2\) to \(y = -x^2\) reflects the graph across the x-axis, so it now opens downwards instead of upwards.
• Reflection across the y-axis: This type of reflection is less common with parabolas, as altering the sign inside the brackets, such as from \(y = (x)^2\) to \(y = (-x)^2\), doesn't affect the standard \(y = x^2\) function significantly in its basic form.

In the given function \(y = -(x+3)^2 - 1\), the negative sign outside the parentheses indicates a reflection across the x-axis. This transformation turns the traditionally upward-opening base parabola downward.

Horizontal Shift

Horizontal shifts move the parabola left or right along the x-axis. This shift is dictated by a constant added to or subtracted from the x-variable. Consider:

• Shift to the left: If the function includes \(x + c\), it signifies a shift of \(c\) units to the left. In our function \(y = -(x + 3)^2 - 1\), the \(+3\) moves the parabola 3 units to the left.
• Shift to the right: Conversely, if the function includes \(x - c\), that denotes a shift of \(c\) units to the right.

This shift directly affects the location of the parabola's vertex. These moves align the vertex of the transformed parabola away from the origin, switching its location along the x-axis.

Vertical Shift

Vertical shifts involve moving the parabola up or down along the y-axis. This shift is achieved by adding or subtracting a constant outside the squared term. Here’s how it works:

• Shift upwards: By adding a constant \(d\), such as turning \(y = x^2\) into \(y = x^2 + d\), the entire graph lifts \(d\) units upwards.
• Shift downwards: Subtracting a number, as in the function \(y = -(x+3)^2 - 1\), pulls the parabola down by one unit from its current position. This is evident in the \(-1\) of the given function, moving the vertex of the parabola down on the y-axis.

Vertical shifts modify the height of the parabola on the plane, affecting only its vertical positioning without altering its width or horizontal placement.