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Chapter 8: Problem 9

For each of the following functions, evaluate \(f(2)\) and \(f(-2)\). a. \(f(x)=x^{2}-5 x-2\) b. \(f(x)=3 x^{2}-x\) c. \(f(x)=-x^{2}+4 x-2\)

Short Answer

Expert verified
a) f(2) = -8, f(-2) = 12; b) f(2) = 10, f(-2) = 14; c) f(2) = 2, f(-2) = -14.

Step by step solution

01

Understanding the Problem

Evaluate the given function at two specific values, namely 2 and -2, for each provided function.
02

Part a: Write Down the Function

Function: \[ f(x) = x^2 - 5x - 2 \]
03

Part a: Evaluate f(2)

Substitute x with 2: \[ f(2) = 2^2 - 5(2) - 2 \]\[ f(2) = 4 - 10 - 2 \]\[ f(2) = -8 \]
04

Part a: Evaluate f(-2)

Substitute x with -2: \[ f(-2) = (-2)^2 - 5(-2) - 2 \]\[ f(-2) = 4 + 10 - 2 \]\[ f(-2) = 12 \]
05

Part b: Write Down the Function

Function: \[ f(x) = 3x^2 - x \]
06

Part b: Evaluate f(2)

Substitute x with 2: \[ f(2) = 3(2^2) - 2 \]\[ f(2) = 3(4) - 2 \]\[ f(2) = 12 - 2 \]\[ f(2) = 10 \]
07

Part b: Evaluate f(-2)

Substitute x with -2: \[ f(-2) = 3(-2)^2 - (-2) \]\[ f(-2) = 3(4) + 2 \]\[ f(-2) = 12 + 2 \]\[ f(-2) = 14 \]
08

Part c: Write Down the Function

Function: \[ f(x) = -x^2 + 4x - 2 \]
09

Part c: Evaluate f(2)

Substitute x with 2: \[ f(2) = -(2^2) + 4(2) - 2 \]\[ f(2) = -4 + 8 - 2 \]\[ f(2) = 2 \]
10

Part c: Evaluate f(-2)

Substitute x with -2: \[ f(-2) = -(-2)^2 + 4(-2) - 2 \]\[ f(-2) = -4 - 8 - 2 \]\[ f(-2) = -14 \]

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Key Concepts

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

quadratic functions
A quadratic function is a type of polynomial that follows the form:
  • \(f(x) = ax^2 + bx + c\).

  • The highest degree term here is \(x^2\), which gives it the name 'quadratic.'

  • The coefficients, \(a\), \(b\), and \(c\) are constants, where \(a e 0\).
These types of functions graph as parabolas. The direction of the parabola ( up or down) depends on the coefficient \(a\):

  • If \(a\) is positive, it opens upwards.

  • If \(a\) is negative, it opens downwards.
Understanding this can be very helpful when evaluating quadratic functions, as it gives insight into the overall behavior and shape of the graph.
substitution method
The substitution method is a basic technique used to evaluate functions, especially quadratics. Here's how it works:
  • You replace all instances of the variable \(x\) with a specific number.
  • You then perform arithmetic operations according to the function's structure to find the output.
Let's consider an example with a quadratic function:
Suppose we have \(f(x) = x^2 - 5x - 2\). To find \(f(2)\), we substitute 2 wherever x appears:
  • \(f(2) = (2)^2 - 5(2) - 2\)
  • Combine like terms: \(4 - 10 - 2\)
  • Thus, \(f(2) = -8\)
This substitution method applies to any function, not only quadratic ones. Each step carefully replaces \(x\) and simplifies.
evaluating functions
Evaluating functions is about finding the output of a function given an input for the variable. The steps involve:
  • Identifying the function
  • Choosing a specific input value for the variable
  • Substituting the chosen value into the function
  • Simplifying the expression to get the output
To evaluate a function, say \(f(x) = -x^2 + 4x - 2\), for \(f(2)\) and \(f(-2)\), follow the steps below:
  • Substitute \(2\) into \(f\): \(f(2) = -(2^2) + 4(2) - 2\)
  • Simplify it: \(-4 + 8 - 2 = 2\)

  • Substitute \(-2\) into \(f\): \(f(-2) = -(-2)^2 + 4(-2) - 2\)
  • Simplify it: \(-4 - 8 - 2 = -14\)
Each of these steps will guide you to correctly evaluate the functions. Consistency and careful substitution are key.

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Most popular questions from this chapter

A typical retirement scheme for state employees is based on three things: age at retirement, highest salary attained, and total years on the job. Annual retirement allowance \(=\) \(\left(\begin{array}{c}\text { total years } \\\ \text { worked }\end{array}\right) \cdot\left(\begin{array}{c}\text { retirement } \\ \text { age factor }\end{array}\right) \cdot\left(\begin{array}{c}\% \text { of highest } \\ \text { salary }\end{array}\right)\) where the maximum percentage is \(80 \%\). The highest salary is typically at retirement. We define: total years worked = retirement age \(-\) starting age retirement age factor \(=0.001\) ?(retirement age -40 ) salary at retirement \(=\) starting salary \(+\) all annual raises a. For an employee who started at age 30 in 1973 with a salary of \(\$ 12,000\) and who worked steadily, receiving a \(\$ 2000\) raise every year, find a formula to express retirement allowance, \(R,\) as a function of employee retirement age, \(A\). b. Graph \(R\) versus \(A\). c. Construct a function \(S\) that shows \(80 \%\) of the employee's salary at age \(A\) and add its graph to the graph of \(R\). From the graph estimate the age at which the employee annual retirement allowance reaches the limit of \(80 \%\) of the highest salary. d. If the rule changes so that instead of highest salary, you use the average of the three highest years of salary, how would your formula for \(R\) as a function of \(A\) change?

Complex number expressions can be simplified by combining the real parts and then the imaginary parts. Add (or subtract) the following complex numbers and then simplify. a. \((4+3 i)+(-5+7 i)\) c. \((7 i-3)+(2-4 i)\) b. \((-2+3 i)-(-3+3 i)\) d. \((7 i-3)-(2-4 i)\)

For each of the following quadratic functions, find the vertex \((h, k)\) and determine if it represents the maximum or minimum of the function. a. \(f(x)=-2(x-3)^{2}+5\) c. \(f(x)=-5(x+4)^{2}-7\) b. \(f(x)=1.6(x+1)^{2}+8\) d. \(f(x)=8(x-2)^{2}-6\)

For each part construct a function that satisfies the given conditions. a. Has a constant rate of increase of $$\$ 15,000 /$$ year b. Is a quadratic that opens upward and has a vertex at (1,-4) c. Is a quadratic that opens downward and the vertex is on the \(x\) -axis d. Is a quadratic with a minimum at the point (10,50) and a stretch factor of 3 e. Is a quadratic with a vertical intercept of (0,3) that is also the vertex

A shot-put athlete releases the shot at a speed of 14 meters per second, at an angle of 45 degrees to the horizontal (ground level). The height \(y\) (in meters above the ground) of the shot is given by the function $$ y=2+x-\frac{1}{20} x^{2} $$ where \(x\) is the horizontal distance the shot has traveled (in meters). a. What was the height of the shot at the moment of release? b. How high is the shot after it has traveled 4 meters horizontally from the release point? 16 meters? c. Find the highest point reached by the shot in its flight. d. Draw a sketch of the height of the shot and indicate how far the shot is from the athlete when it lands.
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