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Chapter 4: Problem 69

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-x^{2}+6 x-1\)

Short Answer

Expert verified
The function has a maximum value of 8.

Step by step solution

01

Identify the form of the quadratic function

The given function is in the form \( f(x) = ax^2 + bx + c \). From the function \( f(x) = -x^2 + 6x - 1\), identify the coefficients: \( a = -1 \), \( b = 6 \), \( c = -1 \).
02

Determine the direction of the parabola

Check the coefficient \( a \). If \( a < 0 \), the parabola opens downward, indicating a maximum value. Here, \( a = -1 \), which is less than 0, so the parabola opens downward and the quadratic function has a maximum value.
03

Find the vertex of the parabola

The x-coordinate of the vertex of a parabola given by \( ax^2 + bx + c \) is found using the formula \( x = \frac{-b}{2a} \). Substituting the values of \( a \) and \( b \), we get: \( x = \frac{-6}{2(-1)} = 3 \).
04

Calculate the function value at the vertex

Substitute \( x = 3 \) into the original function to find the y-coordinate (the maximum value). \( f(3) = -(3)^2 + 6(3) - 1 = -9 + 18 - 1 = 8 \). Therefore, the maximum value of the function is 8.

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Key Concepts

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

Understanding Parabolas
A parabola is a U-shaped curve that is the graph of a quadratic function. Quadratic functions are in the form of \( f(x) = ax^2 + bx + c \). The shape and orientation of a parabola are determined by the coefficient \( a \).
If \( a > 0 \), the parabola opens upwards and has a minimum value. If \( a < 0 \), the parabola opens downwards and has a maximum value. In our example, the quadratic function is \( f(x) = -x^2 + 6x - 1 \), where \( a = -1 \). This negative \'a\' tells us that our parabola opens downward.
Identifying the Vertex
The vertex of a parabola is the highest or lowest point on the graph, depending on the direction it opens. The vertex can be found using the formula \( x = \frac{-b}{2a} \), where \( a \) and \( b \) are coefficients from the quadratic function \( f(x) = ax^2 + bx + c \).
For \( f(x) = -x^2 + 6x - 1 \), we have \( a = -1 \) and \( b = 6 \). Plugging these values into the formula gives us the x-coordinate of the vertex: \( x = \frac{-6}{2(-1)} = 3 \).
This means the vertex occurs at \( x = 3 \). To find the corresponding y-coordinate, we substitute \( x = 3 \) back into the function: \( f(3) = -3^2 + 6(3) - 1 = 8 \). So, the vertex is at (3, 8).
Determining the Maximum Value
Since the parabola opens downward (as \( a = -1 \)), the vertex represents the maximum value of the function. We've already calculated this value when we found the vertex.
Substituting \( x = 3 \) into the function, we get \( f(3) = 8 \). Therefore, the maximum value of the quadratic function \( f(x) = -x^2 + 6x - 1 \) is 8.
This value is crucial, as it shows the highest point that the function can reach.
Understanding Coefficients
Coefficients in a quadratic function \( f(x) = ax^2 + bx + c \) play a vital role in shaping the characteristics of the parabola.
  • The coefficient \( a \) determines the direction of the parabola (upwards if positive, downward if negative).
  • The coefficient \( b \) affects where the vertex is located along the x-axis.
  • The constant \( c \), while not directly influencing the parabola's shape or orientation, adjusts the graph's vertical position.
In our example, \( a = -1 \), \( b = 6 \), and \( c = -1 \).
The negative \( a \) causes the parabola to open downward, while \( b = 6 \) influences where the vertex is positioned.
Using the Quadratic Formula
The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a powerful tool for finding the roots of a quadratic function. The formula gives you the x-values at which the function equals zero. These are the points where the parabola intersects the x-axis.
Although finding the vertex uses a different formula, understanding the quadratic formula is crucial for comprehensive knowledge of quadratic functions.
For the function \( f(x) = -x^2 + 6x - 1 \), the roots could be calculated by plugging \( a = -1 \), \( b = 6 \), and \( c = -1 \) into the quadratic formula. This process reveals the points where the parabola crosses the x-axis.

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Most popular questions from this chapter

(a) Graph fand \(g\) on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-x^{2}+5 x ; \quad g(x)=x^{2}+3 x-4\)

Find the function whose graph is the graph of \(y=\sqrt{x}\) but reflected about the \(y\) -axis.

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-x^{2}+4 x\)

Suppose that \(f(x)=x^{2}+2 x-8\) (a) What is the vertex of \(f ?\) (b) What are the \(x\) -intercepts of the graph of \(f ?\) (c) Solve \(f(x)=-8\) for \(x\). What points are on the graph of \(f ?\) (d) Use the information obtained in parts (a)-(c) to \(\operatorname{graph} f(x)=x^{2}+2 x-8\)

The daily revenue \(R\) achieved by selling \(x\) boxes of candy is \(R(x)=9.5 x-0.04 x^{2}\). The daily cost \(C\) of selling \(x\) boxes of candy is \(C(x)=1.25 x+250 .\) (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as \(P(x)=R(x)-C(x) .\) What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.
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