Question

Graph \(f(x)=(x+1)^{2}-4\) using transformations (shifting, compressing, stretching, and/or reflecting).

Short answer

Shift the graph of \(y = x^2\) left by 1 unit and down by 4 units to obtain \(y = (x+1)^2 - 4\).

Step-by-step solution

Identify the Parent Function

The parent function is the simplest form of the given function that we know how to graph. For the given function, the parent function is \(f(x) = x^2\).

Determine Shifts (Translations)

Next, identify any horizontal or vertical shifts from the parent function. The function \(f(x)=(x+1)^{2}-4\) indicates the following shifts:- Horizontal shift: The term \(x+1\) indicates a shift left by 1 unit.- Vertical shift: The term \(-4\) indicates a shift downward by 4 units.

Apply the Horizontal Shift

Shift the graph of \(y = x^2\) left by 1 unit. This means that every point \((x, y)\) on the graph of \(y = x^2\) will move to \((x-1, y)\).

Apply the Vertical Shift

After shifting left, next shift the graph down by 4 units. This changes every point \((x-1, y)\) on the graph of \(y = (x-1)^2\) to \((x-1, y-4)\).

Plot the Transformed Function

Now plot the transformed function on a coordinate plane. The vertex of the original \(y = x^2\) at \((0,0)\) will now be at \((-1,-4)\). Draw the parabola opening upwards with the new vertex.

Verify by Substitution

To ensure correctness, substitute values for \(x\) into \(f(x)=(x+1)^{2}-4\) and check if the corresponding points lie on the transformed graph. For example, if \(x = 0\): \(f(0) = (0+1)^2 - 4 = 1 - 4 = -3\), so the point \((0, -3)\) should be on the graph.

Key concepts

Parabola

A parabola is a U-shaped curve that you often see in mathematics, especially when dealing with quadratic functions. The general form of a parabola is given by the equation \( y = ax^2 + bx + c \).
In this specific scenario, we're focusing on the simpler form \( y = x^2 \), which is a standard parabola that opens upwards.
The most important feature of a parabola is its vertex, which is the point where the curve changes direction. For the equation \( y = x^2 \), the vertex is at \( (0, 0) \).
Understanding how to manipulate and transform this basic parabola can help you graph more complex functions easily.

Parent Function

The parent function is essentially the simplest form of a given function type that retains the general shape and characteristics you're interested in.
For a parabola, the parent function is \( f(x) = x^2 \). It serves as the baseline or reference point before any transformations like shifting or stretching are applied.
By starting with the parent function, you can understand how different operations change the graph.
In our exercise, \( (x+1)^2 - 4 \) can be broken down starting from the parent function \( x^2 \). This makes it easier to visualize and then apply the necessary transformations step-by-step.

Horizontal Shift

A horizontal shift moves the graph of a function left or right on the coordinate plane.
In the function \( f(x) = (x+1)^2 - 4 \), the term \( (x+1) \) indicates a horizontal shift. Specifically, \( x+1 \) means the graph shifts to the left by 1 unit.
Think of it like this: for each point \( (x, y) \), after the shift it becomes \( (x-1, y) \). This is the opposite of what you might expect since \( +1 \) in the function moves to the left.
So if the vertex of \( y = x^2 \) is at \( (0, 0) \), it would shift to \( (-1, 0) \).

Vertical Shift

A vertical shift moves the graph of a function up or down on the coordinate plane.
In the function \( f(x) = (x+1)^2 - 4 \), the \( -4 \) indicates a vertical shift downward by 4 units.
This means if you have a point \( (x, y) \) on your graph, after the shift, it will be at \( (x, y-4) \).
Combining this with our horizontal shift, if the vertex moved from \( (0, 0) \) to \( (-1, 0) \) because of the horizontal shift, the vertical shift will place it finally at \( (-1, -4) \).
Now, instead of the vertex being at the origin, it's at a new point, and the entire parabola adjusts accordingly to open upwards from this new vertex.