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Chapter 9: Problem 10

Write a quadratic equation of a parabola with \(x\)-intercepts at \(-3\) and 9 and vertex at \((3,-9)\). Express your answer in factored form. (a)

Short Answer

Expert verified
The quadratic equation is \(y = \frac{1}{4}(x + 3)(x - 9)\).

Step by step solution

01

Identify the Form of the Quadratic Equation

The quadratic equation of a parabola with given roots (or \(x\)-intercepts) can be expressed in factored form as \( y = a(x - p)(x - q) \), where \(p\) and \(q\) are the \(x\)-intercepts. Here, \(p = -3\) and \(q = 9\). Thus, the equation starts as \( y = a(x + 3)(x - 9) \).
02

Use the Vertex to Find 'a'

The vertex form of a parabola is useful when finding the coefficient \(a\). The vertex given is \((3, -9)\). Substitute \(x = 3\) and \(y = -9\) into the equation \(y = a(x + 3)(x - 9)\) to solve for \(a\).
03

Substitute the Vertex Coordinates

Substitute \(x = 3\) and \(y = -9\) into the equation: \(-9 = a(3 + 3)(3 - 9)\). Simplify to find \(-9 = a(6)(-6)\).
04

Solve for 'a'

Simplify \(-9 = a(6)(-6)\) to \(-9 = a(-36)\) and solve for \(a\). Divide both sides by \(-36\): \(a = \frac{-9}{-36} = \frac{1}{4}\).
05

Write the Factored Form Equation

Now, substitute \(a = \frac{1}{4}\) back into the factored form equation. The factored form is \(y = \frac{1}{4}(x + 3)(x - 9)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a U-shaped curve that can open upwards or downwards, depending on the sign of a particular coefficient in its equation. It's the graphical representation of a quadratic function, which can be expressed in different algebraic forms. By studying parabolas, we gain insights into their symmetry, vertex, and direction of opening.

Every parabola has certain key components:
  • Vertex: The point where the parabola changes direction, considered its minimum or maximum point depending on the orientation.
  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • X-intercepts: Points where the parabola crosses the x-axis. These are also known as the roots of the function.
Understanding the parabola structure helps in converting quadratic functions to different forms, analysis, and graph predictions.
Factored Form
Factored form is one way to express the equation of a quadratic function. This form usually highlights the x-intercepts or roots of the parabola. A quadratic in factored form looks like this: \[ y = a(x - p)(x - q) \]Here,
  • a is a coefficient that affects the width and direction of the parabola.
  • p and q are the values of the x-intercepts.
Having the function in factored form makes it straightforward to identify the x-intercepts by setting each factor equal to zero. It's especially useful when you want to quickly sketch a graph.

In the given exercise, with x-intercepts at -3 and 9, the factored form of the quadratic equation is expressed as:\[ y = a(x + 3)(x - 9) \]
Vertex Form
The vertex form of a quadratic equation is particularly useful when you want to highlight the vertex of a parabola. This form is given by the equation:\[ y = a(x - h)^2 + k \]In this equation:
  • h and k are the coordinates of the vertex \( h, k \).
  • a remains the coefficient that affects the direction and width of the parabola, just as in the factored form.
Using the vertex form makes it easy to determine how the parabola will stretch or shrink, and whether it opens upwards or downwards. In the context of the exercise, while the solution involves the factored form, one also utilizes the vertex form to find the correct value of a using the given vertex at (3, -9). This conversion is critical to solve real-world problems requiring a vertex to identify or predict a quadratic's behavior.
X-intercepts
X-intercepts, also known as roots or zeros, are points where the graph of a quadratic function crosses the x-axis. These points occur at the solutions to the equation when the quadratic is set equal to zero. Visually, they are the horizontal intersections of the parabola.

To identify the x-intercepts in any quadratic equation:
  • Set the equation equal to zero: \( ax^2 + bx + c = 0 \).
  • Solve for \( x \) using factoring, the quadratic formula, or completing the square.
In the exercise, the x-intercepts are given as -3 and 9. By presenting the quadratic in factored form \( y = (x + 3)(x - 9) \), it's clear that if \( y = 0 \), then \( x \) must be -3 or 9. These points are critical in determining the graph's shape and are essential for plotting the function accurately.

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