Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 14
(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 \(f\) $$f(x)=-x^{2}-6 x$$
Short Answer
Expert verified
Zeros are -6 and 0; the maximum value is 9 at x = -3.
Step by step solution
01
Identify the Quadratic Coefficients
For the function \( f(x) = -x^2 - 6x \), identify the coefficients: \( a = -1 \), \( b = -6 \), and \( c = 0 \).
02
Apply the Quadratic Formula
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the zeros. Plugging in the values, we have \( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(-1)(0)}}{2(-1)} \).
03
Calculate the Discriminant
Calculate the discriminant \( b^2 - 4ac \). Here, it is \( (-6)^2 - 4(-1)(0) = 36 \). Since the discriminant is positive, there are two real roots.
04
Solve for the Zeros
Substitute the discriminant back into the quadratic formula: \( x = \frac{6 \pm \sqrt{36}}{-2} \). Simplify to find \( x = \frac{6 \pm 6}{-2} \).
05
Find Zeros
The two solutions are \( x = \frac{12}{-2} = -6 \) and \( x = \frac{0}{-2} = 0 \). So, the zeros are \( x = -6 \) and \( x = 0 \).
06
Determine the Vertex for Maximum or Minimum
The vertex form of a parabola \( ax^2 + bx + c \) has its vertex at \( x = \frac{-b}{2a} \). Substitute \( a = -1 \) and \( b = -6 \) to find \( x = \frac{-(-6)}{2(-1)} = -3 \).
07
Evaluate the Function at the Vertex
Find the value of \( f(-3) \) by substituting into the function: \( f(-3) = -(-3)^2 - 6(-3) = -9 + 18 = 9 \). The function has a maximum value of 9 at \( x = -3 \).
08
Sketch the Graph
The graph of \( f(x) = -x^2 - 6x \) is a parabola opening downwards, with zeros at \( x = 0 \) and \( x = -6 \) and a vertex at \( (-3, 9) \). This reflects a downward opening parabola with a maximum at the vertex.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
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 roots or zeros of a quadratic function. A quadratic function is generally expressed in the form \( ax^2 + bx + c = 0 \). In our function \( f(x) = -x^2 - 6x \), we have coefficients \( a = -1 \), \( b = -6 \), and \( c = 0 \).
The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's a breakdown of how it works:
For our specific function, substituting in the values for \( a \), \( b \), and \( c \), you can find the zeros using the formula, confirming them as \( x = 0 \) and \( x = -6 \). These are the points where the parabola intersects the x-axis.
The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's a breakdown of how it works:
- \(-b\): This changes the sign of the coefficient \( b \).
- \(\sqrt{b^2 - 4ac}\): The discriminant, determining the nature of the roots.
- The entire formula provides solutions for \( x \), equating to the zeros of the function.
For our specific function, substituting in the values for \( a \), \( b \), and \( c \), you can find the zeros using the formula, confirming them as \( x = 0 \) and \( x = -6 \). These are the points where the parabola intersects the x-axis.
Zeros of a Function
The zeros or roots of a function are the values of \( x \) that make the function equal to zero. Essentially, they are the points where the graph of the function intersects the x-axis.
By using the quadratic formula on our function \( f(x) = -x^2 - 6x \), we calculated the zeros to be \( x = 0 \) and \( x = -6 \). This tells you exactly where the parabola crosses the x-axis.
By using the quadratic formula on our function \( f(x) = -x^2 - 6x \), we calculated the zeros to be \( x = 0 \) and \( x = -6 \). This tells you exactly where the parabola crosses the x-axis.
- Zero at \( x = 0 \)
- Zero at \( x = -6 \)
Vertex of a Parabola
The vertex of a parabola is a crucial point that provides either the maximum or minimum value of the quadratic function, depending on the direction the parabola opens. For a parabola opening downwards, like \( f(x) = -x^2 - 6x \), it is a maximum.
To find the vertex, use the formula for the x-coordinate of the vertex, \( x = \frac{-b}{2a} \). In this case, substituting \( a = -1 \) and \( b = -6 \), we find the vertex's x-coordinate to be \( x = -3 \).
Evaluate the function at \( x = -3 \):\[ f(-3) = -(-3)^2 - 6(-3) = -9 + 18 = 9 \]Therefore, the vertex is \((-3, 9)\). This means at \( x = -3 \), the parabola achieves its maximum value of 9.
To find the vertex, use the formula for the x-coordinate of the vertex, \( x = \frac{-b}{2a} \). In this case, substituting \( a = -1 \) and \( b = -6 \), we find the vertex's x-coordinate to be \( x = -3 \).
Evaluate the function at \( x = -3 \):\[ f(-3) = -(-3)^2 - 6(-3) = -9 + 18 = 9 \]Therefore, the vertex is \((-3, 9)\). This means at \( x = -3 \), the parabola achieves its maximum value of 9.
Maximum and Minimum Values
In quadratic functions, the vertex provides a clue into whether the function has a maximum or minimum value. If \( a \) (from \( ax^2 \)) is positive, the parabola opens upwards, and the vertex is the minimum point. Conversely, if \( a \) is negative, like in this function, the parabola opens downwards, and the vertex is the point of maximum value.
For \( f(x) = -x^2 - 6x \), since \( a = -1 \), the parabola opens downward, indicating a maximum value. We have already established that the maximum value occurs at the vertex \((-3, 9)\), so the maximum value of the function is 9.
For \( f(x) = -x^2 - 6x \), since \( a = -1 \), the parabola opens downward, indicating a maximum value. We have already established that the maximum value occurs at the vertex \((-3, 9)\), so the maximum value of the function is 9.
- The parabola achieves this maximum value at the vertex \( x = -3 \).
- Understanding the direction of the parabola is essential to determining the nature of the vertex.
