Problem 31 - Basic Technical Mathematics with Calculus

Question

Solve the given applied problem. A quadratic equation \(f(x)=0\) has a solution \(x=-1 .\) Its graph has its vertex at (3,4) . What is the other solution?

Step-by-Step Solution

Answer

The other solution is x = 7.

Step 1: Determine the Symmetry Axis

The axis of symmetry of a quadratic equation in the form of a parabola is given by the vertex's x-coordinate. Since the vertex is at (3,4), the axis of symmetry is x = 3. This means that the solutions (roots) must be equidistant from x = 3.

Step 2: Calculate Distance from Vertex to Known Solution

Calculate the distance from the vertex x-coordinate (3) to the known solution (−1). The distance is the absolute difference, |3 - (-1)| = |3 + 1| = 4.

Step 3: Find the Other Solution Using the Symmetry

Since the solutions are symmetric about x = 3, find the other solution by adding the same distance to the axis of symmetry in the positive direction. The other solution is 3 + 4 = 7.

Key Concepts

Axis of Symmetry

The axis of symmetry is a crucial concept in understanding the geometry of a quadratic equation. When a quadratic function is graphed, it forms a U-shaped curve called a parabola. The axis of symmetry is a vertical line that divides the parabola into two identical halves. This line passes through the vertex of the parabola. For a quadratic equation in the standard form, \( ax^2 + bx + c \), the x-coordinate of the axis of symmetry is given by the formula \( x = -\frac{b}{2a} \).
In simpler terms, if you imagine holding a mirror along the axis of symmetry, each half of the parabola would perfectly reflect the other. The axis of symmetry helps us understand the behavior and key properties of the quadratic function.
In our problem, the axis of symmetry is \( x = 3 \), as given by the x-coordinate of the vertex. This tells us the solutions to the quadratic equation are equally spaced from \( x = 3 \). This symmetry allows us to solve for other unknown solutions easily.

Vertex of a Parabola

The vertex of a parabola is a point where the curve changes direction and is either the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. Also, the vertex is often considered the 'tip' of the parabola, where the curve meets the axis of symmetry. Knowing the vertex is vital because it provides valuable information about the entire parabola.
The vertex is denoted as \((h, k)\), where \( h \) is the x-coordinate, and \( k \) is the y-coordinate. For the problem at hand, the vertex is at \((3, 4)\). This means the parabola reaches its minimum or maximum height at the point where x equals 3, and the value at this point is 4.
Identifying the vertex aids in sketching the parabola and understanding the function's extremities, or in this case, helps us identify the axis of symmetry directly.

Distance Calculation in Coordinate Geometry

Distance calculation is a fundamental technique in coordinate geometry, often used to find the length between two points on a coordinate plane. This concept is particularly useful when dealing with geometric properties of functions, such as solving for other roots of a parabola based on symmetry.
To calculate the distance between two x-coordinates, you simply take the absolute difference between them. For instance, in our scenario, the known solution is \( x = -1 \), and the vertex x-coordinate is \( x = 3 \). The distance is calculated as follows:

• Calculate \(|3 - (-1)| = |3 + 1| = 4\).

This gives us the distance from the vertex to the known solution. Since the solutions are symmetric concerning the axis of symmetry \( x = 3 \), this distance helps us find the other solution by moving the same distance in the opposite direction. Thus, the other solution is \( x = 3 + 4 = 7 \).
Understanding distance calculation not only helps in this problem but is also pivotal in broader mathematical contexts.