Problem 19 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=3 x^{2}+1 $$

Step-by-Step Solution

Answer

Vertex: (0, 1), Axis of Symmetry: x = 0, Domain: (-∞, ∞), Range: [1, ∞)

Step 1 - Identify the standard form

The given function is already in the standard form of a parabola, which is \[ f(x) = ax^2 + bx + c \]. Here, \( a = 3 \), \( b = 0 \), and \( c = 1 \).

Step 2 - Find the Vertex

The vertex formula for a parabola in standard form is given by \( x = -\frac{b}{2a} \). Substitute \( a = 3 \) and \( b = 0 \) to get:\[ x = -\frac{0}{2(3)} = 0 \]. Now, substitute \( x = 0 \) back into the function to find the y-coordinate: \[ f(0) = 3(0)^2 + 1 = 1 \]. Thus, the vertex is \( (0, 1) \).

Step 3 - Identify the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. Since the vertex is at \( x = 0 \), the axis of symmetry is \( x = 0 \).

Step 4 - Determine the Domain

The domain of any quadratic function is all real numbers. Therefore, the domain is \( (-\infty, \infty) \).

Step 5 - Determine the Range

Since the parabola opens upwards (because \( a = 3 \) is positive), the lowest point is the y-coordinate of the vertex. Thus, the range is \( [1, \infty) \).

Step 6 - Graph the Parabola

To graph the parabola, plot the vertex at (0, 1) and sketch the curve opening upward. The axis of symmetry is the vertical line at \( x = 0 \). Make sure the curve is symmetric about this line.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point on the graph. It represents the minimum or maximum point, depending on whether the parabola opens upwards or downwards. For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the formula to find the x-coordinate of the vertex is \( x = -\frac{b}{2a} \).
In this example, the quadratic function is \( f(x) = 3x^2 + 1 \), with \( a = 3 \) and \( b = 0 \). Substituting these values into the formula, we get \( x = -\frac{0}{2(3)} = 0 \).
Next, we substitute \( x = 0 \) back into the function to get the y-coordinate: \( f(0) = 3(0)^2 + 1 = 1 \).
Therefore, the vertex for this parabola is \( (0, 1) \).

Axis of Symmetry

The axis of symmetry is an imaginary vertical line that divides the parabola into two mirror-image halves. It passes through the vertex. For a function given by \( f(x) = ax^2 + bx + c \), the axis of symmetry can be found using \( x = -\frac{b}{2a} \).
In our function \( f(x) = 3x^2 + 1 \), we already determined that the vertex is at \( x = 0 \). Hence, the axis of symmetry is simply the line \( x = 0 \).
This line plays an integral part in graphing the parabola, ensuring that the curve opens symmetrically about this vertical line.

Domain of a Function

The domain of a function represents all possible input values (x-values) that the function can accept. For quadratic functions like \( f(x) = ax^2 + bx + c \), the domain is always all real numbers. This is because you can substitute any real number for \( x \) and still get a valid output.
Thus, for our function \( f(x) = 3x^2 + 1 \), the domain is \( (-\infty, \infty) \).
This indicates that the parabola stretches infinitely to the left and right along the x-axis.

Range of a Function

The range of a function is the set of all possible output values (y-values). For a parabola \( f(x) = ax^2 + bx + c \) that opens upwards (when \( a > 0 \)), the y-values start from the vertex and extend to positive infinity. The lowest y-value is the y-coordinate of the vertex.
In our case, the vertex is at \( (0, 1) \), so the y-coordinate is 1. Since the parabola opens upward (indicated by \( a = 3 \), which is positive), the range is \[ [1, \infty) \].
This means the output values start at 1 and go upward indefinitely.

Standard Form of a Quadratic Function

The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \). This form helps in easily identifying the coefficients \( a \), \( b \), and \( c \), which are essential for graphing and analyzing the parabola.
For the function given in this problem, \( f(x) = 3x^2 + 1 \), we see that \( a = 3 \), \( b = 0 \), and \( c = 1 \).
Knowing the standard form allows us to find the vertex, axis of symmetry, and take further steps like determining domain and range effectively. This form is fundamental to understanding and working with quadratic functions.