Problem 25 - Algebra 2 and Trigonometry

Question

In \(19-25,\) express each answer in simplest radical form. Check each answer. A parabola is symmetric under a line reflection. Each real root of the quadratic function \(y=a x^{2}+b x+c\) is the image of the other under a reflection in the axis of symmetry of the parabola. a. What are the coordinates of the points at which the parabola whose equation is \(y=a x^{2}+b x+c\) intersects the \(x\) -axis? b. What are the coordinates of the midpoint of the points whose coordinates were foundin part a? c. What is the equation of the axis of symmetry of the parabola \(y=a x^{2}+b x+c ?\) d. The turning point of a parabola is on the axis of symmetry. What is the \(x\) -coordinate of the turning point of the parabola \(y=a x^{2}+b x+c ?\)

Step-by-Step Solution

Answer

a. \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\); b. \(x = \frac{-b}{2a}\); c. \(x = \frac{-b}{2a}\); d. \(x = \frac{-b}{2a}\).

Step 1: Finding the x-intercepts

The points where a parabola intersects the x-axis are found by solving the equation \( y = ax^2 + bx + c = 0 \). Using the quadratic formula, the solutions are: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. These are the x-coordinates where the parabola intersects the x-axis.

Step 2: Finding the Midpoint of x-intercepts

The midpoint of two points with coordinates \( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) and \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \) is calculated as \( \frac{x_1 + x_2}{2} \). Simplifying: \[ \frac{\left(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\right) + \left(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\right)}{2} = \frac{-2b}{2(2a)} = \frac{-b}{2a} \]. Thus, the x-coordinate of the midpoint is \( \frac{-b}{2a} \).

Step 3: Equation of the Axis of Symmetry

The axis of symmetry of a parabola given by \( y = ax^2 + bx + c \) is a vertical line passing through the midpoint of the x-intercepts. From Step 2, we know the x-coordinate is \( \frac{-b}{2a} \). Therefore, the equation of the axis of symmetry is \( x = \frac{-b}{2a} \).

Step 4: x-coordinate of the Turning Point

The turning point, or vertex, of a parabola lies on the axis of symmetry. Since the axis of symmetry is \( x = \frac{-b}{2a} \), the x-coordinate of the turning point is also \( \frac{-b}{2a} \). This is consistent with the vertex form of a parabola.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function, specifically of degree 2. It is generally expressed in the standard form as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a ot= 0 \). The graph of a quadratic function is a U-shaped curve called a parabola. Parabolas can open upwards or downwards, depending on the sign of \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Quadratic functions have several interesting properties, one of which is symmetry. This means that they reflect evenly across a line, known as the axis of symmetry. Understanding how to manipulate these equations and interpret their graphs is crucial for solving many math problems involving real-world scenarios.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror images. For a quadratic function written as \( y = ax^2 + bx + c \), the equation of the axis of symmetry can be derived from the formula for the x-coordinate of the vertex, which is \( x = \frac{-b}{2a} \). This formula gives the line where the parabola reflects itself evenly.
Understanding the axis of symmetry is crucial because it:

• Helps to find the vertex (or turning point) of the parabola, which is the highest or lowest point depending on the parabola's orientation.
• Is key for solving quadratic equations, particularly when optimizing a quadratic function.

In real-world problems, the axis of symmetry can help in determining maximum or minimum values of the function, providing insights into problem-solving.

x-intercepts

The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. Mathematically, these are the solutions to the equation \( ax^2 + bx + c = 0 \). These can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The term \( \sqrt{b^2 - 4ac} \) inside the quadratic formula is the discriminant. It determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), there are two distinct real x-intercepts.
• If \( b^2 - 4ac = 0 \), there is exactly one real x-intercept (the vertex lies on the x-axis).
• If \( b^2 - 4ac < 0 \), there are no real x-intercepts; the roots are complex numbers.

These intercepts are extremely useful in graphing parabolas and understanding their layout in the coordinate plane.

Turning Point

The turning point of a parabola, also known as the vertex, is the point where the direction of the curve changes. For the quadratic function \( y = ax^2 + bx + c \), the vertex's x-coordinate can be found using \( x = \frac{-b}{2a} \), which aligns with the axis of symmetry. To find the full coordinates of the vertex, substitute this x-value back into the original equation to solve for y.
The turning point is crucial because it:

• Represents the maximum or minimum value of the quadratic function.
• Marks the steepest point on the graph.
• Is used in optimization problems where you need to find the highest or lowest point of certain scenarios.

Visualizing a parabola through its turning point can significantly aid students in understanding the geometry and algebra of quadratic functions.