Problem 21 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ g(x)=(x-1)^{2}-1 $$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (1, -1) and symmetric about the line x=1, opening upwards.

Step 1: Identify the Basic Function

The function given is \( g(x) = (x-1)^2 - 1 \). The basic function here is \( f(x) = x^2 \), which is a standard parabola opening upwards.

Step 2: Determine Horizontal Shift

The expression \((x-1)^2\) indicates a horizontal shift to the right by 1 unit. This is because \(x\) has been replaced by \( x-1 \).

Step 3: Determine Vertical Shift

The whole expression \((x-1)^2 - 1\) shows a downward vertical shift by 1 unit. This is due to the \(-1\) subtracted from \((x-1)^2\).

Step 4: Sketch the Vertex

Combine the shifts to find the vertex of the parabola. The original vertex \((0, 0)\) of \( f(x) = x^2 \) moves to \((1, -1)\) after the shifts.

Step 5: Plot the Parabola Shape

The parabola is symmetric around its vertex at \((1, -1)\), opening upwards. Sketch the parabola maintaining its U-shape, symmetric around the line \(x=1\).

Step 6: Verify with Points

To ensure accuracy, calculate a few additional points. For example, \( g(0) = (0-1)^2 - 1 = 0 \), so plot the point (0, 0). Also, \( g(2) = (2-1)^2 - 1 = 0 \), point (2, 0). These points confirm the correct placement of the parabola.

Key Concepts

Parabola Vertex

In the realm of quadratic functions, the vertex of a parabola is a key feature. The vertex for the quadratic function \( f(x) = x^2 \) is at the origin, which is the point \((0, 0)\). However, transformations such as horizontal and vertical shifts can move this vertex to a different position.
For the function \( g(x) = (x-1)^2 - 1 \), the vertex is derived from both horizontal and vertical shifts. The formula \((x-1)\) suggests a horizontal shift to the right by one unit, while \(-1\) at the end moves the graph downward by one unit.
Therefore, the new vertex of \( g(x) \) is at the point \((1, -1)\). Understanding this allows you to easily sketch where the main focal point of this parabola lies on a graph.

Horizontal Shift

A horizontal shift changes the position of a function graph along the x-axis without altering its shape. In our function \( g(x) = (x-1)^2 - 1 \), the \( (x-1) \) term indicates such a shift.
This change implies that every x-value of the basic parabola \( f(x) = x^2 \) moves right by 1 unit. A positive shift to the right is noted when \( x \) is replaced by \( x-c \) where \( c > 0 \). Conversely, a shift to the left occurs with \( x+c \).

• Right shift: Replace \( x \) with \( x-c \)
• Left shift: Replace \( x \) with \( x+c \)

For our function, the x-values are each moved 1 unit to the right. This helps in redefining the position of all key points, particularly the vertex, on the graph.

Vertical Shift

Vertical shifts adjust the position of a function graph along the y-axis. They can be easily spotted when a constant value is either added or subtracted from the function.
In \( g(x) = (x-1)^2 - 1 \), the \(-1\) at the end represents a vertical shift downward by 1 unit. This affects the entire graph, moving each point down the same vertical distance.

• Upward shift: Add a constant \(+d\)
• Downward shift: Subtract a constant \(-d\)

For our given function, the vertical shift takes the entire parabola down one unit along the y-axis. It repositions the graph's vertex at \((1, -1)\), and ensures that all other points are also adjusted to maintain the correct shape of the parabola.

Function Graph Sketching

Sketching the graph of a quadratic function involves understanding its transformations and plotting them correctly. For \( g(x) = (x-1)^2 - 1 \), start by identifying the basic parabola \( f(x) = x^2 \).
Then apply

• Horizontal shifts: Move right 1 unit.
• Vertical shifts: Move down 1 unit.

This will position the vertex precisely at \((1, -1)\). After identifying the vertex, symmetrically plot the parabola opening upwards. Utilizing points like \( (0, 0) \) and \( (2, 0) \), to further pinpoint accuracy on the graph, helps ensure that the shape remains correct.
Finally, connect these plotted points smoothly, maintaining the upward U-shape, which is characteristic of a parabola. This approach makes visualizing and sketching the graph of quadratic functions, like \( g(x) \), more intuitive and straightforward.