Problem 37 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ x=-\frac{1}{5} y^{2}+2 y-4 $$

Step-by-Step Solution

Answer

Vertex: (2, -4), Axis of symmetry: \( x = 2 \, Domain: \( (-\text{∞}, \text{∞}) \), Range: \( [-4, \text{∞}) \)

Step 1 - Identify the vertex form

Identify the function's form. The given function is in the vertex form:\[ f(x) = a(x-h)^2 + k \]where \(a = 2, h = 2, k = -4\)

Step 2 - Determine the vertex

From the vertex form, the vertex \( (h, k) \) can be extracted:Vertex: (2, -4)

Step 3 - Find the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex. It has the equation \( x = h \):Axis of symmetry: \( x = 2 \)

Step 4 - Identify the domain

The domain of a parabola that is a quadratic function such as this one is all real numbers:Domain: \( (-\text{∞}, \text{∞}) \)

Step 5 - Determine the range

Since \( a = 2 \) is positive, the parabola opens upwards. The range starts from the vertex's y-value and goes to \text{∞}:Range: \( [-4, \text{∞}) \)

Step 6 - Plotting the graph

To graph the parabola:1. Plot the vertex at (2, -4).2. Draw the axis of symmetry, \( x = 2 \).3. Use additional points on either side of the axis of symmetry to get the shape, e.g., \( x = 1 \) and \( x = 3 \).4. Sketch the parabola, ensuring it opens upwards.

Key Concepts

vertex form

The vertex form of a quadratic function makes it very easy to identify the graph's vertex. This form is given by ewline f(x) = a(x-h)^2 + k.ewline Here,

• a represents the vertical stretch or compression.
• h is the x-coordinate of the vertex.
• k is the y-coordinate of the vertex.

By comparing the function f(x) = 2(x-2)^2 -4, we find ewline a = 2, ewline h = 2,ewline k = -4. So, the vertex is at (2, -4). This point is the lowest point of the parabola since 'a' is positive.ewline Using the vertex form also helps in sketching the graph quickly.

axis of symmetry

The axis of symmetry is a line that divides the parabola into two mirror images. For a quadratic function in vertex form f(x) = a(x-h)^2 + k, the axis of symmetry is always x = h.ewline For our function f(x) = 2(x-2)^2 -4, the axis of symmetry is x = 2. This vertical line passes through the vertex and ensures the parabola looks the same on both sides of this line.

domain and range

Understanding the domain and range of a quadratic function helps describe all possible values of x (domain) and f(x) (range).ewline Domain:

• For any quadratic function, the domain is always all real numbers since we can plug any x-value into the function and compute an output.
• Therefore, the domain of f(x) = 2(x-2)^2 - 4 is (-∞, ∞).

Range:

• The range relates to the output values, or f(x).
• Since the parabola opens upwards (because a > 0), the minimum point of the parabola is the vertex. Thus, the range starts from the y-coordinate of the vertex and goes to positive infinity.
• Therefore, the range of f(x) = 2(x-2)^2 - 4 is [-4, ∞).

quadratic functions

Quadratic functions form the core of parabolas in algebra. A quadratic function typically looks like f(x) = ax^2 + bx + c, where a, b, and c are constants.

• If the coefficient 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.
• The vertex formula for converting to vertex form provides an efficient way to graph these functions. Our example function, f(x) = 2(x-2)^2 -4, is already in vertex form.
• To graph, find the vertex, plot it, draw the axis of symmetry, add additional points around the vertex, and sketch the curve.

By understanding the quadratic functions and their components, it's easier to interpret and plot their graphs.