Problem 96 - Algebra and Trigonometry

Question

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$ f(x)=(x-3)^{2}+2 ; \quad(6,11) $$

Step-by-Step Solution

Answer

The axis of symmetry of the given parabola is x = 3. Using this, a second point on the parabola with the same y-coordinate as the given point (6, 11) is (0, 11).

Step 1: Find the axis of symmetry

The axis of symmetry of the parabola \( f(x) = (x - h)^2 + k \) is \( x = h \). Looking at the given function \( f(x) = (x - 3)^2 + 2 \), we can identify \( h = 3 \). Therefore, the axis of symmetry is \( x = 3 \).

Step 2: Use the axis of symmetry to find a second point

The given point is (6, 11). The x-coordinate 6 is 3 units to the right of the axis of symmetry (3). Therefore, to get the x-coordinate of the new point, we have to move 3 units to the left of the axis of symmetry. That gives us an x-coordinate of 0. Since it was mentioned that the y-coordinate remains the same, the new point on the parabola will be (0, 11).

Key Concepts

Parabola

A parabola is a U-shaped curve that appears in many different areas of mathematics. You often encounter parabolas when dealing with quadratic functions, where the graph of a quadratic equation forms a parabola. This curve is symmetric, meaning it mirrors itself perfectly along a vertical line, known as the axis of symmetry.

Parabolas can open upwards or downwards, and the direction they open depends on the sign of the coefficient in front of the squared term in the equation. If it's positive, the parabola opens upwards; if negative, downwards. Understanding this characteristic is crucial for identifying key features such as the vertex, the axis of symmetry, and the focus of the parabola.

Quadratic Function

A quadratic function is a polynomial function of the form \[f(x) = ax^2 + bx + c\] where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These functions always produce a parabola when graphed. Every quadratic function has a vertex, which is either the highest or lowest point on the graph.

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

The standard form of the quadratic equation, which is useful for finding the vertex and the axis of symmetry, is given as: \[ f(x) = (x-h)^2 + k \] Here, \(h\) and \(k\) are the coordinates of the vertex of the parabola, with the axis of symmetry given by \(x = h\). These components are critical for understanding the geometric properties of the quadratic function.

Coordinates

Coordinates are a set of values used to determine the exact position of a point on a graph. They are written in the form \((x, y)\), where \(x\) is the position along the horizontal axis and \(y\) is the position along the vertical axis. In the context of parabolas and quadratic functions, coordinates are essential for identifying points on the curve.

In the exercise given, you determined one point as \((6, 11)\). Understanding that the parabola is symmetric allows you to apply the axis of symmetry to find another point with the same \(y\)-coordinate: \((0, 11)\). This makes it evident how coordinates help in visualizing and manipulating points on a graph.

Symmetry

Symmetry is a key mathematical concept that simplifies graphing and problem-solving. In terms of parabolas, symmetry means that each side of the parabola is a mirror image of the other across the axis of symmetry.

• The axis of symmetry can be easily found from the quadratic function's formula, often noted as \(x = h\), where \(h\) is derived from the expression in the vertex form \((x-h)^2\).
• Knowing the axis of symmetry helps in identifying other critical points on the parabola, making calculations more intuitive and precise.

For instance, with the axis of symmetry \(x = 3\), you move in equal units from this line to determine symmetric points. This quality of parabolas being symmetric is a powerful tool in both calculus and algebra for simplifying complex equations and drawing graphs efficiently.