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Chapter 2: Problem 45

Find the roots and sketch the graphs of the quadratic functions:\(-2 x^{2}-3 x+2\)

Short Answer

Expert verified
The roots are \( x = -2 \) and \( x = \frac{1}{2} \). The graph is an upside-down parabola intersecting the x-axis at these points.

Step by step solution

01

Identify the Quadratic Equation

The quadratic equation given is \( -2x^2 - 3x + 2 \). A standard quadratic equation is in the form \( ax^2 + bx + c \). Here, \( a = -2 \), \( b = -3 \), and \( c = 2 \).
02

Use the Quadratic Formula

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In this case, substitute \( a = -2 \), \( b = -3 \), and \( c = 2 \) into the formula.
03

Calculate the Discriminant

The discriminant is \( b^2 - 4ac \). Substitute \( b = -3 \), \( a = -2 \), and \( c = 2 \) into the formula to get \( (-3)^2 - 4(-2)(2) = 9 + 16 = 25 \).
04

Compute the Roots

Substitute the discriminant back into the quadratic formula to find the roots:\[ x = \frac{-(-3) \pm \sqrt{25}}{2(-2)} = \frac{3 \pm 5}{-4} \]Calculate the two roots:- \( x_1 = \frac{3 + 5}{-4} = \frac{8}{-4} = -2 \)- \( x_2 = \frac{3 - 5}{-4} = \frac{-2}{-4} = \frac{1}{2} \)
05

Sketch the Graph of the Quadratic Function

The graph of the quadratic function is a parabola. Since \( a = -2 \) is negative, the parabola opens downward. The roots of the function are at \( x = -2 \) and \( x = \frac{1}{2} \), so these are the points where the parabola intersects the x-axis. The vertex can be found using \( x = -\frac{b}{2a} = -\frac{-3}{2(-2)} = \frac{3}{4} \). Substitute \( x = \frac{3}{4} \) back into the function to find the y-coordinate of the vertex. The vertex is a maximum point due to the negative leading coefficient.

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Key Concepts

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

Quadratic Equation
A quadratic equation is a fundamental concept in algebra and appears in various fields of science and engineering. It's an equation of the second degree, which means it includes a term squared (^2). The general form is:
  • \( ax^2 + bx + c = 0 \)
where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The values of \( a \), \( b \), and \( c \) influence the shape and position of the parabola when the equation is graphed. Understanding these components:
  • \( a \) determines if the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).
  • \( b \) and \( c \) influence the position of the vertex and the axis of symmetry of the parabola.
In the exercise, our quadratic equation is \(-2x^2 - 3x + 2\), identifying \( a = -2 \), \( b = -3 \), and \( c = 2 \). A negative \( a \) indicates a downward-opening parabola.
Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of a quadratic equation. These roots are the solutions where the quadratic intersects the x-axis. The formula is:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula might seem complex at first, but it systematically finds both potential solutions鈥攁lso known as roots鈥攆or any quadratic equation. Every component in this formula has a specific role:
  • \( -b \) determines the horizontal shift due to the linear portion.
  • \( \pm \sqrt{b^2 - 4ac} \) gives us the potential for two solutions, accounting for both positive and negative square roots.
  • \( 2a \) calculates the average effects of the parabola's shape around the vertex.
In the provided exercise, substituting \( a = -2 \), \( b = -3 \), and \( c = 2 \) into the equation helps find where the quadratic function \(-2x^2 - 3x + 2\) hits the x-axis, yielding the roots \(-2\) and \(\frac{1}{2}\). These are the x-values where the function equals zero.
Graphing Parabolas
Graphing a parabola, which is the graphical representation of a quadratic function, gives us a visual understanding of its behavior. A parabola is always symmetric around its vertex, which acts as a central balance point. For quadratic functions like our example \(-2x^2 - 3x + 2\):
  • The vertex is located by computing \( x = -\frac{b}{2a} \).
  • For this equation, the vertex happens at \( x = \frac{3}{4} \).
After finding \( x \), plug it back into the original equation to find the corresponding \( y \)-value. This vertex is a peak since the parabola opens downward (negative \( a \)). Visually:
  • If \( a > 0 \), the parabola opens upwards with a vertex at the lowest point.
  • If \( a < 0 \), the parabola opens downward, making the vertex a peak.
The x-intercepts or roots are \(-2\) and \(\frac{1}{2}\), and they mark where the parabola crosses the x-axis. Understanding this sketch helps you see both the direction and the intersection points of the curve.
Discriminant Calculation
The discriminant is an essential part of the quadratic formula, found under the radical in \( \sqrt{b^2 - 4ac} \) and determines the nature of the roots of the quadratic equation:
  • If \( b^2 - 4ac > 0 \), there are two distinct real roots.
  • If \( b^2 - 4ac = 0 \), there is one real root (a double root).
  • If \( b^2 - 4ac < 0 \), the roots are complex or imaginary and not real numbers.
In our quadratic equation \(-2x^2 - 3x + 2\):
  • The discriminant is calculated as \( (-3)^2 - 4(-2)(2) = 9 + 16 = 25 \).
Since 25 is greater than zero, it indicates there are two real and distinct roots. This aligns with our earlier calculations of roots \(-2\) and \(\frac{1}{2}\). Discriminant calculations help predict and understand the nature of solutions before solving the equation.

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