Problem 172 - Elementary Algebra

Question

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=-2 x^{2}-8 x-3 $$

Step-by-Step Solution

Answer

Axis of symmetry: x = -2. Vertex: (-2, 5).

Step 1: Identify coefficients

Given the quadratic equation in the form: \[ y = ax^2 + bx + c \]Identify the coefficients: \( a = -2 \), \( b = -8 \), and \( c = -3 \).

Step 2: Find the axis of symmetry

The formula for the axis of symmetry for a quadratic equation is: \[ x = \frac{-b}{2a} \]Plug in the coefficients \( a = -2 \) and \( b = -8 \): \[ x = \frac{-(-8)}{2(-2)} = \frac{8}{-4} = -2 \]Thus, the axis of symmetry is \( x = -2 \).

Step 3: Calculate the vertex

To find the vertex, substitute the axis of symmetry \( x = -2 \) back into the original equation. \[ y = -2(-2)^2 - 8(-2) - 3 \]Simplify the expression: \[ y = -2(4) + 16 - 3 \]\[ y = -8 + 16 - 3 \]\[ y = 5 \]Therefore, the vertex is at \( (-2, 5) \).

Key Concepts

Axis of Symmetry

The axis of symmetry in a parabola is an important concept. It is a vertical line that passes through the highest or lowest point of the parabola, known as the vertex.
The axis of symmetry divides the parabola into two mirror-image halves. You can think of it as the 'line of balance' for the parabola. In mathematical terms, for any quadratic equation of the form:
\[ y = ax^2 + bx + c \]
The formula to find the axis of symmetry is:
\[ x = \frac{-b}{2a} \].

By substituting the given coefficients into the formula, we get:
\[ x = \frac{-(-8)}{2(-2)} = \frac{8}{-4} = -2 \].

Therefore, the axis of symmetry for the given quadratic equation is \( x = -2 \).

Vertex

The vertex of a parabola is its highest or lowest point. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, the vertex is the maximum point.
To find the vertex of a quadratic equation in the form:
\[ y = ax^2 + bx + c \],
we need to know two things: the x-coordinate and the y-coordinate of the vertex. We've already determined that the x-coordinate is the axis of symmetry. For our example, that value is -2.

To find the y-coordinate, we substitute \( x = -2 \) back into the original quadratic equation:
\[ y = -2(-2)^2 - 8(-2) - 3 \]
First, calculate \( (-2)^2 \) which equals 4. Then:
\[ y = -2(4) + 16 - 3 \]
\[ y = -8 + 16 - 3 \]
\[ y = 5 \].

So, the vertex of the parabola is at point (-2, 5).

Quadratic Equation

Quadratic equations are mathematical expressions of the form:
\[ y = ax^2 + bx + c \],
where:

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

These equations represent parabolas when graphed. The shape and direction of the parabola depend on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards and has a minimum point (the vertex).
• If \( a < 0 \), the parabola opens downwards and has a maximum point.

Understanding how to find the axis of symmetry and vertex is crucial because they provide us with key information about the parabola's graph. They help us understand the location and shape of the curve.

For our example:
\[ y = -2x^2 - 8x - 3 \]
we saw how the coefficients \( a = -2 \), \( b = -8 \), and \( c = -3 \) were used to find the axis of symmetry and vertex, thus graphing an accurate parabola.