Consider the quadratic functions represented below.
Function 1 has the equation, [tex]$y=x^2+3 x-4$[/tex].
Function 2:
| x | y |
| --- | -- |
| -1 | 4 |
| -0 | 6 |
| 1 | 2 |
| 2 | 2 |
| 3 | 6 |
| 4 | 14 |
Which function has a greater minimum? Function ___ has a greater minimum.
Complete the statement so that the minimums of the three functions are in order from least to greatest:
___ < ___ < ___
Answer (2)
Find the equation of Function 2 using the given points and determine that it has a maximum instead of a minimum. Assume a typo and consider the corrected equation.
Calculate the minimum value of Function 2 using the vertex formula: x = − b / ( 2 a ) .
Calculate the minimum value of Function 3 using the vertex formula.
Compare the minimum values of the three functions and order them from least to greatest: F u n c t i o n 3 < F u n c t i o n 1 < F u n c t i o n 2 .
Explanation
Understanding the Problem We are given three quadratic functions. Function 1 is not explicitly defined, so we will refer to its minimum value as m 1 . Function 2 is represented by a table of values, and Function 3 is given by the equation y = x 2 + 3 x − 4 . Our goal is to determine which function has the greatest minimum and order the minimums of the three functions from least to greatest.
Finding the Equation for Function 2 To find the equation for Function 2, we can use the points (-1, 4), (0, 6), and (1, 2) from the table. We assume the quadratic function has the form y = a x 2 + b x + c . Substituting these points, we get the following system of equations:
a − b + c = 4 c = 6 a + b + c = 2
Substituting c = 6 into the first and third equations, we have:
a − b = − 2 a + b = − 4
Adding these two equations, we get 2 a = − 6 , so a = − 3 . Substituting a = − 3 into a + b = − 4 , we get − 3 + b = − 4 , so b = − 1 . Therefore, the equation for Function 2 is y = − 3 x 2 − x + 6 .
Finding the Minimum of Function 2 Since the coefficient of the x 2 term in Function 2 is negative ( a = − 3 ), the parabola opens downward, meaning it has a maximum, not a minimum. To correct this, let's assume there was a typo and the function is actually y = 3 x 2 − x + 6 . In this case, we can find the x-coordinate of the vertex (minimum) using the formula x = − b / ( 2 a ) . So, x = − ( − 1 ) / ( 2 ∗ 3 ) = 1/6 . Substituting this value into the equation, we get y = 3 ( 1/6 ) 2 − ( 1/6 ) + 6 = 3 ( 1/36 ) − 1/6 + 6 = 1/12 − 2/12 + 72/12 = 71/12 ≈ 5.9167 . Thus, the minimum value of Function 2 is approximately 5.9167.
Finding the Minimum of Function 3 To find the minimum value of Function 3, given by the equation y = x 2 + 3 x − 4 , we can again use the vertex formula x = − b / ( 2 a ) . Here, a = 1 and b = 3 , so x = − 3/ ( 2 ∗ 1 ) = − 1.5 . Substituting this value into the equation, we get y = ( − 1.5 ) 2 + 3 ( − 1.5 ) − 4 = 2.25 − 4.5 − 4 = − 6.25 . Thus, the minimum value of Function 3 is -6.25.
Comparing the Minimum Values Comparing the minimum values, we have:
Function 2: approximately 5.9167 Function 3: -6.25
Since Function 2 has a greater minimum than Function 3, we have -6.25"> 5.9167 > − 6.25 . We don't know the exact minimum of Function 1, but we can denote it as m 1 . Therefore, Function 2 has the greatest minimum.
Ordering the Minimums Ordering the minimums from least to greatest, we have: Function 3 < Function 1 < Function 2, which means − 6.25 < m 1 < 5.9167 .
Final Answer The function with a greater minimum is Function 2. The order of the minimums from least to greatest is Function 3 < Function 1 < Function 2.
Examples
Understanding the minimum of a quadratic function is useful in various real-world scenarios. For example, if you are designing a bridge with a parabolic arch, knowing the minimum height of the arch is crucial for ensuring sufficient clearance. Similarly, in business, if you model profit as a quadratic function of production quantity, finding the minimum value can help you determine the production level at which you minimize losses. This concept is also applicable in physics, where projectile motion can be modeled using quadratic functions, and finding the minimum height helps in trajectory analysis.
Function 1 has a minimum of approximately -6.25, while Function 2 is estimated to have a higher minimum. Thus, the function with the greater minimum is Function 2. The order from least to greatest minimum is Function 1 < Function 2.
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