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Chapter 2: Problem 47

Find the roots and sketch the graphs of the quadratic functions:\(-x^{2}+6 x-9\)

Short Answer

Expert verified
The root is x = 3, the parabola opens downwards, vertex at (3, 0).

Step by step solution

01

Identify the Quadratic Function

The given quadratic function is \[-x^2 + 6x - 9\]This function is in the standard form \( ax^2 + bx + c \) where \( a = -1 \), \( b = 6 \), and \( c = -9 \).
02

Use the Quadratic Formula

To find the roots of the quadratic function, we can use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute \(a = -1\), \(b = 6\), and \(c = -9\) into the formula.
03

Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \):\[6^2 - 4(-1)(-9) = 36 - 36 = 0\].
04

Solve for x Using the Quadratic Formula Result

Since the discriminant is 0, there is one real root. Substitute the values back into the quadratic formula:\[x = \frac{-6 \pm \sqrt{0}}{-2}\]The single solution is\[x = \frac{6}{2} = 3\].
05

Sketch the Graph

Plot the vertex of the parabola at the point \((3, 0)\), because for a given quadratic expression with one root, this root is also the vertex of the parabola. Since \(a = -1\), the parabola opens downwards. The graph is symmetric about the line \(x = 3\) and has no other x-intercepts.

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Key Concepts

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

Understanding the Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation. A quadratic equation is typically in the format of:
  • \( ax^2 + bx + c = 0 \)
In this formula, \(a\), \(b\), and \(c\) are constants. To solve for the roots, you use the quadratic formula:
  • \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here's the step-by-step breakdown:
  • Identify the coefficients: First, match the equation to the quadratic form and identify \(a\), \(b\), and \(c\).
  • Calculate the discriminant \(b^2 - 4ac\): This helps to determine the number of roots.
  • Solve for \(x\): Plug the values into the formula to find the roots.
This formula helps simplify and systematize finding the roots of any quadratic function.
The Role of the Discriminant
The discriminant is the part of the quadratic formula contained within the square root: \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the quadratic equation.
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the quadratic equation has exactly one real root, or a repeated root. This was the case in the given problem leading to the single root \(x = 3\).
  • If the discriminant is negative, there are no real roots, but two complex roots.
In the given exercise, the discriminant was zero, indicating that our quadratic has exactly one repeated real solution. This unique property often implies that the vertex of the parabola sits on the x-axis at this root.
Visualizing Quadratic Equations with Parabolas
A quadratic equation, when graphed, creates a curve called a parabola. Understanding the characteristics of a parabola will help in sketching it:
  • When the coefficient \(a\) is positive, the parabola opens upwards, looking like a smile.
  • If \(a\) is negative, as in the given problem, the parabola opens downwards, appearing like a frown.
Key features of a parabola include:
  • Vertex: The highest or lowest point of the parabola. For \(ax^2 + bx + c\) with \(x = -\frac{b}{2a}\) as the x-coordinate at the vertex. For our exercise, the vertex is at \((3,0)\).
  • Axis of symmetry: A vertical line through the vertex; it divides the parabola into two mirror images. In our example, this is the line \(x = 3\).
  • Intercepts: Parabolas might cross the x-axis (the roots) and always cross the y-axis. For problems with a single root, like ours, the parabola touches the x-axis only at its vertex.
By understanding these characteristics, it becomes easier to sketch and analyze quadratic functions visually.

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