Problem 55 - Algebra for College Students

Question

Give a step-by-step explanation of how to find the vertex of the parabola determined by the equation \(f(x)=\) \(-x^{2}-6 x-5\)

Step-by-Step Solution

Answer

The vertex of the parabola is \((-3, 4)\).

Step 1: Identify the Standard Form of a Parabola

The standard form of a parabola is given by the equation \(f(x) = ax^2 + bx + c\). In our problem, the equation is \(f(x) = -x^2 - 6x - 5\), where \(a = -1\), \(b = -6\), and \(c = -5\). This identification is crucial for finding the vertex.

Step 2: Recall the Vertex Formula

The vertex of a parabola \(ax^2 + bx + c\) can be found using the vertex formula for the x-coordinate: \(x = \frac{-b}{2a}\). This formula helps us find the x-coordinate of the vertex.

Step 3: Calculate the x-coordinate of the Vertex

Substitute the values of \(a\) and \(b\) into the vertex formula. Here, \(a = -1\) and \(b = -6\), so \(x = \frac{-(-6)}{2(-1)} = \frac{6}{-2} = -3\). Thus, the x-coordinate of the vertex is \(-3\).

Step 4: Calculate the y-coordinate of the Vertex

To find the y-coordinate, substitute \(x = -3\) back into the original equation \(f(x) = -x^2 - 6x - 5\). Calculate: \(f(-3) = -(-3)^2 - 6(-3) - 5 = -9 + 18 - 5 = 4\). So, the y-coordinate of the vertex is \(4\).

Step 5: Write the Coordinates of the Vertex

The vertex is the point \((x, y)\). From the previous steps, we have found that the vertex is \((-3, 4)\).

Key Concepts

Standard Form of a Parabola

The standard form of a parabola provides a convenient starting point to identify essential features like the vertex. Parabolas can generally be expressed by the equation:

• \(f(x) = ax^2 + bx + c\)

Here, \(a\), \(b\), and \(c\) are constants, and each has its role:

• \(a\): Determines the direction of the parabola. If \(a > 0\), the parabola opens upwards. If \(a < 0\), the parabola opens downwards.
• \(b\): Influences the horizontal placement and shape of the graph.
• \(c\): Represents the y-intercept, the point where the parabola crosses the y-axis.

In our example, the parabola given by \(f(x) = -x^2 - 6x - 5\) is in standard form with:

• \(a = -1\)
• \(b = -6\)
• \(c = -5\)

This form is important because it allows us to apply the vertex formula directly to find the vertex of the parabola.

Vertex Formula

The vertex formula is a critical tool for finding the vertex of a parabola. Given a quadratic equation in the standard form \(ax^2 + bx + c\), the vertex formula helps find where the turning point or vertex occurs.For the x-coordinate of the vertex, use the formula:

• \(x = \frac{-b}{2a}\)

This formula comes from the process of completing the square, a method for rewriting quadratic expressions in a different form. It efficiently gives the x-coordinate without needing more complex algebra.Using this formula makes finding the vertex straightforward, as long as the coefficients \(a\), \(b\), and \(c\) are identified correctly from the standard form.

x-coordinate of a Vertex

Finding the x-coordinate of the vertex is straightforward using the vertex formula. By identifying the values of \(a\) and \(b\) in the standard form of the quadratic equation, you plug these into the formula \(x = \frac{-b}{2a}\).For our specific example, where the equation is \(f(x) = -x^2 - 6x - 5\), the coefficients are:

• \(a = -1\)
• \(b = -6\)

Plug these into the formula:

• \(x = \frac{-(-6)}{2(-1)} = \frac{6}{-2} = -3\)

Hence, the x-coordinate of the vertex is \(-3\), pinpointing the horizontal position where the parabola pivots.

y-coordinate of a Vertex

Once the x-coordinate of the vertex has been determined, finding the y-coordinate is the next step. This involves substituting the x-value back into the original quadratic equation and solving for \(f(x)\).For the example equation \(f(x) = -x^2 - 6x - 5\), and with the x-coordinate \(x = -3\), substitute and solve:

• \(f(-3) = -(-3)^2 - 6(-3) - 5\)
• \(= -9 + 18 - 5\)
• \(= 4\)

Thus, the y-coordinate of the vertex is \(4\). The vertex, being a significant point of the parabola, is then fully identified with coordinates \((-3, 4)\). Knowing both coordinates provides a complete understanding of where the parabola changes direction, offering insights into its graphical representation.