Problem 40 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ G(x)=3 x^{2}+1 $$

Step-by-Step Solution

Answer

Vertex: (0, 1); Axis of Symmetry: x = 0.

Step 1: Identify the Vertex

The given quadratic function is in the form \(G(x) = ax^2 + c\), where \(a = 3\) and \(c = 1\). Since there is no linear term (no \(bx\) term where \(b = 0\)), the vertex is at \(x = 0\). Calculate the \(y\)-coordinate by plugging \(x = 0\) into \(G(x)\): \(G(0) = 3(0)^2 + 1 = 1\). Thus, the vertex is \((0, 1)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function in the form \(ax^2 + bx + c\) is given by \(x = -\frac{b}{2a}\). Here, \(b = 0\), thus the axis of symmetry is \(x = 0\).

Step 3: Plot the Graph and Label

Draw the graph of \(G(x) = 3x^2 + 1\) by plotting the vertex at \((0, 1)\). The parabola opens upward since \(a = 3 > 0\). The axis of symmetry is the vertical line \(x = 0\). Sketch the parabola such that it is symmetric about this line.

Key Concepts

Vertex of a Quadratic

The vertex of a quadratic function is a crucial point on its graph. It represents the peak or the lowest point of the parabola, depending on whether it opens upwards or downwards. For a quadratic function expressed in the standard form:

• \(f(x) = ax^2 + bx + c\)

The vertex can be found using the formula \(x = -\frac{b}{2a}\). This formula helps determine the x-coordinate of the vertex.
For our particular example, we have \(G(x) = 3x^2 + 1\), a case where there is no linear "bx" term.

• Since \(b = 0\), the vertex is located directly at \(x = 0\).
• To find the y-coordinate, substitute back into the quadratic function:\(G(0) = 3(0)^2 + 1 = 1\).
• Thus, the vertex is at the point \((0, 1)\).

The vertex is significant because it provides valuable information about the graph, including whether it is a minimum or maximum point. This, in turn, helps in sketching the graph accurately.

Axis of Symmetry

The axis of symmetry is an invisible vertical line that divides the parabola into two mirror-image halves. For any quadratic function written in the form:

• \(f(x) = ax^2 + bx + c\), the equation of the axis of symmetry is \(x = -\frac{b}{2a}\).

This line passes through the vertex of the parabola.
In our provided example, with \(G(x) = 3x^2 + 1\):

• The value of \(b\) is 0, resulting in the axis of symmetry being \(x = 0\).
• Therefore, the parabola is symmetrical about the y-axis.

The axis of symmetry is beneficial for plotting the graph as it ensures the parabola maintains its symmetrical property, providing a guide for how the parabola curves around this line.

Graphing Parabolas

Graphing a quadratic function involves a few key steps that help ensure clarity and accuracy in representing the parabola:

• Identify the vertex and use it as a reference point for plotting.
• Locate the axis of symmetry to understand how the parabola opens and curves.
• Consider the direction of the parabola's opening: if \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.

For \(G(x) = 3x^2 + 1\):

• The vertex is at \((0, 1)\), which is the lowest point since \(a = 3 > 0\), indicating it opens upwards.
• The axis of symmetry at \(x = 0\) acts as a guideline for plotting both sides of the graph equally from the center.

When graphing, start by plotting the vertex, then draw the axis of symmetry. Sketch the parabola with emphasis on its upward opening, making sure the curves on either side reflect the symmetry about the axis. Graphing becomes straightforward when these elements are clearly understood.