Problem 33 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ x=(y+2)^{2}+1 $$

Step-by-Step Solution

Answer

Vertex: (-2, -1), Axis of Symmetry: x = -2, Domain: (-∞, ∞), Range: [-1, ∞)

Step 1: Identify the Parabola Form

The function is given as \(f(x)=(x+2)^{2}-1\). This is in vertex form \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.

Step 2: Determine the Vertex

Compare \(f(x)=(x+2)^{2}-1\) with the vertex form. Here, \(a = 1\), \(h = -2\), \(k = -1\). Thus, the vertex is \((-2, -1)\).

Step 3: Identify the Axis of Symmetry

The axis of symmetry for a parabola in vertex form \((x-h)^2\) is the line x = h. Thus, the axis of symmetry is \(x = -2\).

Step 4: Define the Domain

The domain of any quadratic function is all real numbers. This can be written as \((-fty, fty)\).

Step 5: Determine the Range

Since the parabola opens upwards (as a > 0), the range is all real numbers greater than or equal to the y-coordinate of the vertex. Therefore, the range is \([-1, fty)\).

Step 6: Graph the Parabola

Plot the vertex \((-2,-1)\). Draw the axis of symmetry at \(x=-2\). Since \(a=1\), the parabola opens upwards. Sketch the graph accordingly.

Key Concepts

vertex form

Parabolas are often expressed in vertex form, making it easier to identify key features of their graph. Vertex form is given by the equation \(f(x) = a(x-h)^2 + k\), where \(h\) and \(k\) are constants. This form provides straightforward information about the vertex of the parabola, which is the point \((h, k)\). The constant \(a\) influences the width and direction of the parabola. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. Vertex form simplifies understanding the transformation applied to the standard parabola \(y = x^2\). For example, in the equation \(f(x) = (x + 2)^2 - 1\), we identify \(h = -2\), \(k = -1\), and \(a = 1\). This tells us the vertex is at \((-2, -1)\), and the parabola opens upwards.

vertex

The vertex of a parabola is a crucial feature, as it represents the highest or lowest point depending on the direction the parabola opens. In vertex form \(f(x) = a(x-h)^2 + k\), the vertex is given by the coordinates \((h, k)\). This point is derived directly from the constants in the equation. For instance, in the provided function \(f(x) = (x + 2)^2 - 1\), comparing it to the vertex form reveals \(h = -2\) and \(k = -1\), making the vertex \((-2, -1)\). Understand that the vertex is where the parabola changes direction, which is vital for graphing and analyzing the function.

axis of symmetry

An essential feature of parabolas is the axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For a function in vertex form \(f(x) = a(x-h)^2 + k\), the axis of symmetry can be easily identified as the line \(x = h\). For our function \(f(x) = (x + 2)^2 - 1\), with \(h = -2\), the axis of symmetry is \(x = -2\). This line helps in sketching the parabola accurately, ensuring that each side is symmetrical relative to this axis.

domain and range

Understanding the domain and range of a function is key. The domain of a quadratic function is always all real numbers, as a quadratic equation is defined for any real value of \(x\). This is written as \((-\infty, \infty)\).
The range of the function \(f(x) = a(x-h)^2 + k\) depends on the value of \(k\) and the direction in which the parabola opens. If \(a > 0\), the parabola opens upwards, and the range is \([k, \infty)\). If \(a < 0\), the parabola opens downwards, and the range is \((-\infty, k]\).
In our example \(f(x) = (x + 2)^2 - 1\), since \(a = 1\) (positive value), the parabola opens upwards with a vertex at \((-2, -1)\). Hence, the range is \([-1, \infty)\).