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Chapter 2: Problem 15

Each of the graphs of the functions has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if \(f(x)=a x^{2}+b x+c,\) then \(f(x)\) has a relative minimum point when \(a>0\) and a relative maximum point when \(a<0.1\) $$f(x)=-x^{2}-8 x-10$$

Short Answer

Expert verified
The graph of \ f(x) = -x^2 - 8x - 10 \ has a vertex at \ (-4, 6)\, representing a relative maximum point. It opens downward.

Step by step solution

01

Identify the quadratic function

The given quadratic function is \(f(x) = -x^2 - 8x - 10 \). Since the coefficient of \(x^2\) is negative (-1), the function opens downwards, indicating a relative maximum point.
02

Find the vertex

The vertex of a quadratic function \(f(x) = ax^2 + bx + c\) where \(a < 0\) can be found using the formula for the x-coordinate of the vertex, \(-\frac{b}{2a} = -\frac{-8}{2(-1)} = 4\). Therefore, the x-coordinate of the vertex is \(-4\). Substitute back to the function to find the y-coordinate: \ f(-4) = -(-4)^2 - 8(-4) -10 = -16 + 32 -10 = 6\.
03

Vertex and concavity

The vertex is at \((-4, 6)\) and since \(a = -1 \ < 0\), the concavity of the function is downward (relative maximum).
04

Sketch the graph

Plot the vertex \((-4, 6)\) on the coordinate plane. Since the function opens downward, sketch a parabola with its highest point at \((-4, 6)\) and the arms moving downward to indicate a maximum point.

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Key Concepts

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

Relative Maximum
In the study of quadratic functions, the term 'relative maximum' is highly significant. Here, we're dealing with a function of the form \(f(x) = -x^2 - 8x - 10\). The most important feature of this function is the coefficient of \(x^2\), which is negative (-1). When this coefficient is negative, it tells us that the parabola opens downwards. A downwards-opening parabola reaches its peak at the vertex, meaning it attains its highest value there. This highest value is called the relative maximum. In this example, the value at the vertex \((-4, 6)\) is the relative maximum. To summarize:
  • For a quadratic function with a negative \(a\), the function has a relative maximum.
This helps in understanding the behavior and shape of the graph quickly.
Vertex of Parabola
The vertex is an essential concept in graphing and analyzing quadratic functions. The vertex represents the turning point of a parabola. For the quadratic function \(f(x) = -x^2 - 8x - 10\), we can find the vertex by using the formula for the x-coordinate vertex, \(-\frac{b}{2a}\):
\(-\frac{-8}{2(-1)} = -4\).
Once we have the x-coordinate, we substitute it back into the function to find the y-coordinate:
\(f(-4) = -(-4)^2 - 8(-4) - 10 = -16 + 32 - 10 = 6\).
This calculation tells us the vertex is at \((-4, 6)\). Here's a quick summary to remember:
  • The x-coordinate of the vertex is found using \(-\frac{b}{2a}\).
  • Substituting this x-coordinate back into the function gives the y-coordinate.
  • The vertex is a crucial point which can be a maximum or minimum based on the sign of \(a\).
Knowing the vertex allows you to sketch the graph accurately and understand its maximum or minimum points.
Concavity
Concavity describes the direction in which a parabola opens. Understanding concavity is crucial for visualizing and sketching quadratic functions. For the quadratic function \(f(x) = -x^2 - 8x - 10\), the coefficient of \(x^2\) is \(a = -1\). When \(a\) is less than 0, the parabola opens downward. This means the graph curves downward from the vertex. To clarify:
  • When \(a > 0\), the parabola opens upward, and the vertex is a relative minimum.
  • When \(a < 0\), the parabola opens downward, and the vertex is a relative maximum.
For our function, since \(a\) is negative, the parabola's concavity is downward. This tells us that the graph must peak at the vertex, forming a peak at \((-4, 6)\) before heading downwards on either side. Being aware of concavity helps sketch the graph effectively and predict its behavior.

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Most popular questions from this chapter

The graph of each function has one relative extreme point. Find it (giving both \(x\) - and \(y\) -coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. $$f(x)=\frac{1}{4} x^{2}-2 x+7$$

Each of the graphs of the functions has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. $$f(x)=\frac{1}{3} x^{3}+2 x^{2}-5 x+\frac{8}{3}$$

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