Question Video: Using a Variation Function to Find an Unknown Mathematics

If the variation function 𝑓(𝑥) = 𝑎𝑥² + 𝑏𝑥 at 𝑥 = 𝑑 is 𝑉(ℎ) = 𝑎ℎ² + 𝑏ℎ, what is the value of 𝑑?

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Video Transcript

If the variation function 𝑓 of 𝑥 is equal to 𝑎𝑥 squared plus 𝑏𝑥 at 𝑥 is equal to 𝑑 is given by 𝑉 of ℎ is equal to 𝑎ℎ squared plus 𝑏ℎ, what is the value of 𝑑?

In this question, we’re given a quadratic function 𝑓 of 𝑥 which is equal to 𝑎𝑥 squared plus 𝑏𝑥. And we’re told when 𝑥 is equal to 𝑑, the variation function of this quadratic function 𝑓 of 𝑥 is given by 𝑉 of ℎ is equal to 𝑎ℎ squared plus 𝑏ℎ for some unknown constants 𝑎 and 𝑏. We need to use this information to determine the value of 𝑑.

To do this, let’s start by recalling what we mean by the variation function of a function 𝑓 of 𝑥 at 𝑥 is equal to 𝑑. We can recall it’s the function 𝑉 of ℎ which is equal to 𝑓 evaluated at 𝑑 plus ℎ minus 𝑓 evaluated at 𝑑. In other words, this function measures how much the function 𝑓 changes when its input values change from 𝑑 to 𝑑 plus ℎ. And in this question, we’re given an expression for the variation function 𝑉 of ℎ. It’s 𝑎ℎ squared plus 𝑏ℎ.

We can substitute this expression into the left-hand side of the equation. And we can also find an expression for the right-hand side of this equation by noting we’re given the function 𝑓 of 𝑥. It’s 𝑎𝑥 squared plus 𝑏𝑥. So we can substitute 𝑥 is equal to 𝑑 plus ℎ and 𝑥 is equal to 𝑑 into this function to find an expression for the right-hand side of this equation. First, if we substitute 𝑥 is equal to 𝑑 plus ℎ into the function 𝑓 of 𝑥, we get 𝑎 times 𝑑 plus ℎ all squared plus 𝑏 multiplied by 𝑑 plus ℎ. Then, we need to subtract 𝑓 evaluated at 𝑑, which we can find by substituting 𝑥 is equal to 𝑑 into 𝑓 of 𝑥. We subtract 𝑎𝑑 squared plus 𝑏𝑑. This then gives us an equation in terms of 𝑎, ℎ, 𝑏, and 𝑑.

Let’s now rearrange this equation. Let’s start by distributing the exponent over the parentheses. And we can do this by using the binomial formula or by expanding the brackets 𝑑 plus ℎ multiplied by 𝑑 plus ℎ. We get 𝑑 squared plus two 𝑑ℎ plus ℎ squared. And we need to multiply this expression by 𝑎. We can also distribute the negative over the two terms at the end of this expression. This gives us that 𝑎ℎ squared plus 𝑏ℎ is equal to 𝑎 times 𝑑 squared plus two 𝑑ℎ plus ℎ squared plus 𝑏 times 𝑑 plus ℎ minus 𝑎𝑑 squared minus 𝑏𝑑.

We can simplify the right-hand side further by distributing the factor of 𝑎 and the factor of 𝑏 over the parentheses. Doing this gives us the following equation, which we can simplify. We have 𝑎𝑑 squared minus 𝑎𝑑 squared, which is equal to zero. And we also have 𝑏 times 𝑑 minus 𝑏 times 𝑑, which is equal to zero. We can also note that we have 𝑎ℎ squared on both the left- and right-hand side of the equation and 𝑏 times ℎ on both the left- and right-hand side of the equation. So we can subtract 𝑎ℎ squared from both sides of the equation and 𝑏 plus ℎ from both sides of the equation. This gives us that zero is equal to two times 𝑎 times 𝑑 multiplied by ℎ.

So, for this equation to hold true, one of our unknowns must be equal to zero. It’s worth noting this can’t be ℎ because ℎ is a variable. It can take any value for our variation function. And 𝑎 is a parameter of our original equation; it’s not equal to zero. Otherwise, our original function would not be quadratic. So the only way for this equation to hold true for all values of ℎ would be for 𝑑 to be equal to zero. Therefore, we were able to show if the variation function of 𝑓 of 𝑥 is equal to 𝑎𝑥 squared plus 𝑏𝑥 at 𝑥 is equal to 𝑑 is 𝑉 of ℎ is equal to 𝑎ℎ squared plus 𝑏ℎ, then the value of 𝑑 must be zero.