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Chapter 2: Problem 21

(a) Use the quadratic formula to find the zeros of \(f .\) (b) Find the maximum or minimum value of \(f(x)\) (c) Sketch the graph of \(f\) $$f(x)=-2 x^{2}+16 x-26$$

Short Answer

Expert verified
The zeros are \(2 - \sqrt{3}\) and \(2 + \sqrt{3}\). The maximum value is 6 at \(x = 4\).

Step by step solution

01

Identify coefficients for the quadratic formula

The quadratic function is given by \( f(x) = -2x^2 + 16x - 26 \). To apply the quadratic formula, identify the coefficients: \( a = -2 \), \( b = 16 \), and \( c = -26 \).
02

Apply the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients: \( x = \frac{-16 \pm \sqrt{16^2 - 4(-2)(-26)}}{2(-2)} \).
03

Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \): \( 16^2 - 4(-2)(-26) = 256 - 208 = 48 \). The positive discriminant means there are two real roots.
04

Solve for the roots

Substituting the discriminant back into the quadratic formula gives: \( x = \frac{-16 \pm \sqrt{48}}{-4} \). Simplifying further, \( x = \frac{-16 \pm 4\sqrt{3}}{-4} \), yielding the roots \( x = 2 - \sqrt{3} \) and \( x = 2 + \sqrt{3} \).
05

Find the vertex

Since the quadratic function opens downward (\( a = -2 < 0 \)), it has a maximum point. The vertex \( x \)-coordinate is \( x = \frac{-b}{2a} = \frac{-16}{2(-2)} = 4 \).
06

Calculate the maximum value

Substitute \( x = 4 \) back into the function to find the maximum value: \( f(4) = -2(4)^2 + 16(4) - 26 = -32 + 64 - 26 = 6 \). The maximum value of the function is 6.
07

Sketch the graph

The graph is a downward-opening parabola with its vertex at (4, 6), crossing the x-axis at approximately \( x = 2 - \sqrt{3} \) and \( x = 2 + \sqrt{3} \). It is symmetric about \( x = 4 \).

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool used to find the solutions or "roots" of a quadratic equation of the form \( ax^2 + bx + c = 0 \). These roots are the values of \( x \) for which the quadratic function equals zero. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how you can use it:
  • Identify the coefficients \( a \), \( b \), and \( c \) in the quadratic equation.
  • Plug these values into the quadratic formula.
  • Calculate the discriminant \( b^2 - 4ac \) inside the square root.
  • Use the plus-minus symbol (\( \pm \)) to find the two possible roots for \( x \).
In our example, the quadratic formula helps us find that the zeros for \( f(x) = -2x^2 + 16x - 26 \) are approximately \( x = 2 - \sqrt{3} \) and \( x = 2 + \sqrt{3} \). These solutions tell us where the parabola intersects the x-axis.
Vertex of a Parabola
The vertex of a parabola is its highest or lowest point, depending on whether the parabola opens upwards or downwards. For a quadratic equation in the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:\[x = \frac{-b}{2a}\]Substitute this value back into the equation to find the y-coordinate. The vertex provides crucial information:
  • If \( a > 0 \), the parabola opens upwards, and the vertex is a minimum point.
  • If \( a < 0 \), like in our example with \( a = -2 \), the parabola opens downwards, making the vertex a maximum point.
In the case of \( f(x) = -2x^2 + 16x - 26 \), the vertex is calculated to be at \( (4, 6) \). This is the maximum value of the function, as the parabola opens downward.
Discriminant
The discriminant is part of the quadratic formula and is found inside the square root: \( b^2 - 4ac \). It tells us about the nature and number of roots a quadratic equation has:
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there's exactly one real root, meaning the parabola touches the x-axis at one point.
  • If the discriminant is negative, there are no real roots, and the parabola does not intersect the x-axis.
In our function \( f(x) = -2x^2 + 16x - 26 \), the discriminant is 48. Being positive, it confirms the existence of two real roots, which are where the parabola crosses the x-axis.
Graphing Quadratic Equations
To graph a quadratic equation, you need to find several key features of the function: roots, vertex, and the direction (upward or downward) in which the parabola opens.
  • Begin by finding the roots using the quadratic formula. These will be your x-intercepts.
  • Locate the vertex using the formula for the x-coordinate, and substitute to find the y-coordinate. This point is either the peak or the valley of the parabola.
  • Determine the parabola's direction based on the sign of \( a \). If \( a > 0 \), the parabola opens upward. If \( a < 0 \), it opens downward.
  • Plot these points on the coordinate graph and use symmetry about the vertex to draw the parabola.
For \( f(x) = -2x^2 + 16x - 26 \), the graph is a downward-opening parabola with its vertex at (4, 6). It crosses the x-axis at approximately \( x = 2 - \sqrt{3} \) and \( x = 2 + \sqrt{3} \). The symmetry around \( x = 4 \) helps complete the curve.

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