Problem 39 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=4 x^{2}+\frac{1}{4} x-8 $$

Step-by-Step Solution

Answer

The graph of the function opens upwards. The coordinates of the vertex are \((-1/32, -8.0039)\). The equation of the axis of symmetry is \(x = -1/32\).

Step 1: Determine the Opening Direction of the Graph

Look at the leading coefficient \(a\) in the quadratic equation. Since \(a = 4\) which is greater than 0, the graph of the function opens upwards.

Step 2: Determine the Vertex

The x-coordinate of the vertex is given by \(-b/2a\). Here, \(b = 1/4\) and \(a = 4\). Hence, the x-coordinate of the vertex is \(- (1/4) / (2 * 4) = -1/32\). To find the y-coordinate, substitute \(x = -1/32\) into the equation to get \(y = 4 (-1/32)^{2} + 1/4 * (-1/32) - 8\). Hence the vertex is at \((-1/32, -8.0039)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry is the vertical line through the vertex. Since the vertex has x-coordinate \(-1/32\), the equation of the axis of symmetry is \(x = -1/32\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a significant point that represents the peak of the graph. Whether the parabola opens up or down, the vertex is either the highest or the lowest point on the graph.

To find the vertex, you can use the formula for the x-coordinate, \(-\frac{b}{2a}\), where \(b\) and \(a\) come from the quadratic equation in the form \(ax^2 + bx + c\).

In our exercise, for the equation \(y = 4x^2 + \frac{1}{4}x - 8\):

• \(a = 4\)
• \(b = \frac{1}{4}\)

Substituting into the formula gives:
\(x = -\frac{\frac{1}{4}}{2 \times 4} = -\frac{1}{32}\)

Once we have \(x\), we plug it back into the equation to find \(y\). Thus, the vertex is at \((-\frac{1}{32}, -8.0039)\).

Knowing the vertex helps us understand the shape and position of the parabola in a graph.

Axis of Symmetry

The axis of symmetry is an essential feature of parabolas, as it gives the line that divides the graph into two mirror images.

In any quadratic function, this line runs vertically through the vertex. It is crucial because:

• It shows balance, reflecting the parabola symmetrically on both sides.
• It acts as a guide to finding other points on the graph.

The equation for the axis of symmetry is simply \(x = h\), where \(h\) is the x-coordinate of the vertex. From our exercise, we know:

• The vertex \(x\)-coordinate is \(-\frac{1}{32}\)

Therefore, the axis of symmetry is \(x = -\frac{1}{32}\).

This line is like a mirror, giving you a way to understand how the parabola will look and behave.

Leading Coefficient

The leading coefficient in a quadratic equation plays a crucial role in determining the direction the parabola opens, as well as its width.

The general form of a quadratic equation is \(ax^2 + bx + c\), where \(a\) is the leading coefficient.

For our equation \(y = 4x^2 + \frac{1}{4}x - 8\):

• The leading coefficient \(a = 4\)

This value can tell us:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.
• Larger values of \(|a|\) make the parabola narrow.
• Smaller values of \(|a|\) make it wider.

Since \(a = 4\) is positive, this parabola opens upwards, and being larger than 1, it is relatively narrow.

Understanding \(a\) helps you predict the parabola's appearance and direction.