Problem 23 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=3 x^{2} $$

Step-by-Step Solution

Answer

Vertex: (0, 0), Axis of Symmetry: x = 0, Domain: (-∞, ∞), Range: [0, ∞).

Step 1: Identify the form of the quadratic equation

The given function is in the form of a standard quadratic equation: y = ax^2 + bx + cFor this specific exercise: f(x) = 3x^2This implies that a = 3, b = 0, and c = 0.

Step 2: Find the vertex

For a parabola of the form y = ax^2 + bx + cthe vertex (h, k) can be found using the formulas: h = -b/(2a), k = f(h).Since b = 0 and a = 3, we get: h = -0/(2 * 3) = 0.Now calculate k: k = f(0) = 3(0)^2 = 0.Thus, the vertex is (0, 0).

Step 3: Determine the axis of symmetry

The axis of symmetry for the parabola can be given by the line x = h.Since the vertex is (0, 0), the axis of symmetry is x = 0.

Step 4: Calculate the domain

The domain of any quadratic function is all real numbers, written as: Domain: (-∞, ∞).

Step 5: Determine the range

Since the parabola opens upwards (a = 3 > 0), the range includes all y-values greater than or equal to the y-coordinate of the vertex. Range: [0, ∞).

Step 6: Graph the parabola

To graph the parabola, plot the vertex at (0, 0). Given that a = 3, the parabola will be narrower than the basic parabola y = x^2. Draw a symmetric curve opening upwards from the vertex.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept as it represents the highest or lowest point on the graph. For a quadratic function in the form: \(y = ax^2 + bx + c\), the vertex can be found using the formulas:

• \(h = -\frac{b}{2a}\)
• \(k = f(h)\)

In this specific exercise, our quadratic function is \(f(x) = 3x^2\). Here, \(a = 3\), \(b = 0\), and \(c = 0\). Calculating \(h\) gives us:\(h = -\frac{0}{2 \times 3} = 0\).For \(k\), we plug in the value of \(h\) into the function: \(k = f(0) = 3(0)^2 = 0\). Thus, the vertex of the parabola is at the point: (0, 0). This point is where the parabola changes direction.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. For any quadratic function described by \(y = ax^2 + bx + c\), the axis of symmetry can be calculated as:\[ x = -\frac{b}{2a} \] In our function \(f(x) = 3x^2\), the vertex is at (0, 0), so the axis of symmetry is straightforward: \(x = 0\). This means the parabola is symmetric about the y-axis. Whenever you have the vertex, you automatically know the axis of symmetry.

Domain and Range of Functions

Understanding the domain and range helps to know the limits within which the function operates. The domain refers to all possible x-values that can be input into the function. For any quadratic function, including \(f(x) = 3x^2\), the domain is all real numbers:\[ \text{Domain: } (-\infty, \infty)\] The range, however, is the set of possible y-values that the function can output. Since our parabola opens upwards and the vertex is at (0, 0), the smallest y-value is 0. So, the range is:\[ \text{Range: } [0, \infty) \] This tells us the function outputs values starting from 0 and going to positive infinity, but never any negative numbers.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form: \(y = ax^2 + bx + c\). These equations form parabolas when graphed. Some key features include:

• The vertex
• Axis of symmetry
• Direction (upwards if \(a > 0\), downwards if \(a < 0\))
• Domain and range

For our function \(f(x) = 3x^2\), because \(a = 3\), the parabola opens upwards and is narrower compared to the standard parabola \(y = x^2\). These characteristics help in sketching the graph and understanding the behavior of quadratic functions.