Problem 27 - Precalculus : Building Concepts and Connections

Question

Use transformations to graph the quadratic function and find the vertex of the associated parabola. $$f(x)=(x+2)^{2}-1$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = (x+2)^2 - 1\) is a parabola that opens upwards and has its vertex at \((-2, -1)\). It is the graph of the base function \(f(x) = x^{2}\) shifted two units to the left and one unit down.

Step 1: Identify the base function and the transformations

The base function is \(f(x) = x^{2}\). The transformations that have been applied are a horizontal shift two units to the left, represented by \( (x+2)^{2} \), and a vertical shift one unit down, represented by \(-1\)

Step 2: Graph the function with transformations

The graph of the function \(f(x) = x^{2}\) is a parabola that opens upwards. Now we are going to apply the transformations. A horizontal shift two units to the left moves every point of the graph two units to the left. A vertical shift one unit down moves every point of the graph one unit down.

Step 3: Find the vertex of the parabola

The vertex form of a parabola is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. In the given function, \(f(x) = (x+2)^2 - 1\), the vertex \((h, k)\) can be found by equating \(x+2\) to 0 (to find \(h\)) , and observing the constant term which is \(-1\) (this is \(k\)). So, the vertex is \((-2, -1)\).

Key Concepts

Parabolas

Parabolas are a type of curve on a graph that represent the graph of a quadratic function. You'll often see them in the shape of a "U" or flipped upside down, depending on how they open. Parabolas arise from quadratic equations, which are polynomial functions of degree two, typically written in the form \(f(x) = ax^2 + bx + c\), where "\(a\)," "\(b\)," and "\(c\)" are constants.

The graph of a basic quadratic function \(f(x) = x^2\) is a simple parabola that opens upwards with its vertex at the origin \((0,0)\). The vertex is the lowest or highest point of the parabola, depending on whether it opens upwards or downwards.

• When "\(a\)" is positive, the parabola opens upwards.
• When "\(a\)" is negative, the parabola opens downwards.

The width of the parabola can also be affected by the value of "\(a\)." A larger absolute value of "\(a\)" makes the parabola "narrower," while a smaller absolute value makes it "wider." Understanding how parabolas work is crucial for graphing quadratic functions and interpreting their transformations.

Graph Transformations

Graph transformations change the position, shape, or size of the graph on a coordinate plane. These can occur through several processes, including translations, reflections, stretches, and compressions.

In quadratic functions, graph transformations are often easier to identify and implement. Here, we'll focus on translations, which consist of shifts along the axes.

• Horizontal shifts move the graph left or right. A function \(f(x) = (x + k)^2\) represents a shift \(k\) units to the left if \(k\) is positive, or to the right if \(k\) is negative.
• Vertical shifts move the graph up or down. Adding or subtracting a constant \(c\) to \(f(x)\), as in \(f(x) = x^2 + c\), results in moving the graph up by \(c\) units if \(c\) is positive, and down by \(c\) units if \(c\) is negative.

In the exercise, the function \(f(x) = (x+2)^2 - 1\) includes a horizontal shift two units to the left and a vertical shift one unit down. Understanding how these shifts work helps in accurately graphing the function and determining the new vertex position.

Vertex Form

The vertex form of a quadratic function is particularly useful for easily identifying the vertex of a parabola. This form is written as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

To convert a standard form quadratic function to vertex form, you can complete the square or recognize transformations directly from the equation. This method greatly simplifies the graphing process since the vertex gives a clear reference for drawing the parabola.

• In the vertex form, "\(h\)" gives the horizontal shift. If "\(h\)" comes from \((x-h)\), remember that you'll set \(x-h = 0\) to find the "\(h\)" value directly.
• "\(k\)" indicates the vertical shift, which is simply the constant added or subtracted at the end of the function.

When given \(f(x) = (x+2)^2 - 1\), the function is already in vertex form. Identifying \(h = -2\) (as the graph shifts left, making it negative) and \(k = -1\) provides a vertex at \((-2, -1)\). Knowing the vertex form allows you to quickly find the key details of a parabola's graph without extensive calculations.