The Minimum Value of a Quadratic Function: A Question of Symmetry

  • MHB
  • Thread starter Monoxdifly
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In summary, a quadratic function with a symmetrical axis of x = a has a minimum value of -p, where p is not equal to 0. The expression a – f(a) equals 0, and if the symmetrical axis is x = a, then the minimum value is also a – f(a). However, if the question was asking for a + f(a), then the answer would be -4.
  • #1
Monoxdifly
MHB
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A quadratic function \(\displaystyle f(x)=x^2+2px+p\) has the minimum value of –p with \(\displaystyle p\neq0\). If the curve's symmetrical axis is x = a, then a – f(a) = ...
A. –6
B. –4
C. 4
D. 6
E. 8

Because the curve's symmetrical axis is x = a, then:
\(\displaystyle -\frac{2p}{2(1)}=a\)
–p = a

a – f(a) = –p + (–p) = 0

I got zero. Is there anything I did wrong?
 
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  • #2
Monoxdifly said:
$a – f(a) = –p + (–p) = 0

$a - f(a) = {\color{red}-p - (-p)} = -p + p = 0$

agree with zero ... maybe recheck the original problem statement?
 
  • #3
If the problem is indeed asking for $a - f(a)$ then the answer is zero. However ...

The minimum value occurs on the axis of symmetry. Therefore the minimum value is $f(a) = f(-p) = (-p)^2 + 2p(-p) + p = p-p^2$. But you are told that the minimum value is $-p$. Therefore $p-p^2 = -p$, and since $p\ne0$ it follows that $p=2$. Hence $a = -2$, and $f(a)$ is also $-2$. Therefore $a-f(a) = 0$, as we already knew. BUT, if the quetion was actually asking for $a\;{\color{red}+}\,f(a)$ then that would be $-2-2 = -4$, which has the advantage of being one of the multiple choices.

So I agree with skeeter that you should recheck the original problem statement, and in particular look again at whether it is actually asking for $a+f(a)$.
 
  • #4
I checked the problem and it said a – f(a), so probably the writer didn't press the Shift button correctly when he/she intended to type "+".
 

Related to The Minimum Value of a Quadratic Function: A Question of Symmetry

1. What is the meaning of "a - f(a)" in a scientific context?

In a scientific context, "a - f(a)" typically represents the difference between a variable "a" and its corresponding function "f(a)". It is used to calculate the change in the value of "f(a)" as "a" changes.

2. How is "a - f(a)" related to the concept of derivatives?

"a - f(a)" is closely related to the concept of derivatives in calculus. It can be thought of as the derivative of the function "f(a)" with respect to the variable "a". This means that it represents the rate of change of "f(a)" as "a" changes.

3. Can "a - f(a)" be negative?

Yes, "a - f(a)" can be negative. This would indicate that the value of "f(a)" is greater than the value of "a". In other words, the function "f(a)" is decreasing as "a" increases.

4. How is "a - f(a)" used in real-world applications?

"a - f(a)" is used in a variety of real-world applications, particularly in fields such as economics, physics, and engineering. It is commonly used to analyze and predict changes in quantities such as stock prices, velocity, and voltage.

5. Is "a - f(a)" the same as "f(a) - a"?

No, "a - f(a)" is not the same as "f(a) - a". The order of the variables in a subtraction operation changes the result. In this case, "a - f(a)" represents the difference between "a" and "f(a)", while "f(a) - a" represents the difference between "f(a)" and "a".

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