Problem 70 - Elementary Algebra

Question

Problem: The vertex of a parabola that opens up is \((4,-5)\). Write the equation of the axis of symmetry. Incorrect

Step-by-Step Solution

Answer

The equation of the axis of symmetry is \( x = 4 \).

Step 1: Identify the Vertex

The vertex is given as \( (4, -5) \). The key information needed is the x-coordinate of the vertex, which is 4.

Step 2: Understand the Axis of Symmetry

For a parabola that opens up or down, the axis of symmetry is the vertical line that passes through the vertex. This line can be represented as x = h, where h is the x-coordinate of the vertex.

Step 3: Write the Equation

Since the x-coordinate of the vertex is 4, the equation of the axis of symmetry is \( x = 4 \).

Key Concepts

vertex of a parabola

The vertex of a parabola is an important point that determines the shape and position of the parabola. In a quadratic equation of the form \(y = ax^2 + bx + c\), the vertex can be found using the formula for the x-coordinate \(x = -\frac{b}{2a}\).

The vertex represents the highest or lowest point of the parabola:

• For parabolas that open upwards, the vertex is the minimum point.
• For parabolas that open downwards, the vertex is the maximum point.

The given problem states the vertex as \( (4, -5) \), which means the parabola has its minimum (since it opens up) at this point.

Understanding the vertex helps in determining the general shape and direction of the parabola.

x-coordinate

The x-coordinate is the horizontal value in a pair of coordinates. For the vertex of a parabola \( (h, k) \), \ 'h' \ represents the x-coordinate.

In the context of the vertex of a parabola, the x-coordinate is crucial for determining the axis of symmetry. The given problem provides an x-coordinate of \ '4' \ for the vertex \( (4, -5) \).
The visual representation is known as the midpoint that the parabola curves around.

Once the x-coordinate of the vertex is known, it helps to create the equation of the axis of symmetry easily. Just remember, the x-coordinate of the vertex is always the value used in the axis of symmetry formula.

vertical line equation

The axis of symmetry of a parabola is represented by a vertical line. Vertical lines are unique because they go straight up and down rather than diagonally or horizontally.

The general equation for a vertical line is \ x = \ where \ \ denotes a specific x-value.
Given the vertex \ (4, -5) \, the x-coordinate is \ '4' \.

Therefore, the equation of the axis of symmetry will be:
\[ x = 4 \]This vertical line indicates that for every point on the axis of symmetry, the x-value remains \ '4' \.
Knowing how to write and use the vertical line equation aids in graphing the parabola and understanding its directional symmetry.