The portal has been deactivated. Please contact your portal admin.

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

If the variation function 𝑓(π‘₯) = π‘Žπ‘₯Β² + 𝑏π‘₯ at π‘₯ = 𝑑 is 𝑉(β„Ž) = π‘Žβ„ŽΒ² + π‘β„Ž, what is the value of 𝑑?

03:54

Video Transcript

If the variation function 𝑓 of π‘₯ is equal to π‘Žπ‘₯ squared plus 𝑏π‘₯ at π‘₯ is equal to 𝑑 is given by 𝑉 of β„Ž is equal to π‘Žβ„Ž squared plus π‘β„Ž, what is the value of 𝑑?

In this question, we’re given a quadratic function 𝑓 of π‘₯ which is equal to π‘Žπ‘₯ squared plus 𝑏π‘₯. And we’re told when π‘₯ is equal to 𝑑, the variation function of this quadratic function 𝑓 of π‘₯ is given by 𝑉 of β„Ž is equal to π‘Žβ„Ž squared plus π‘β„Ž for some unknown constants π‘Ž 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 π‘₯ at π‘₯ is equal to 𝑑. We can recall it’s the function 𝑉 of β„Ž which is equal to 𝑓 evaluated at 𝑑 plus β„Ž 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 β„Ž. And in this question, we’re given an expression for the variation function 𝑉 of β„Ž. It’s π‘Žβ„Ž squared plus π‘β„Ž.

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 π‘₯. It’s π‘Žπ‘₯ squared plus 𝑏π‘₯. So we can substitute π‘₯ is equal to 𝑑 plus β„Ž and π‘₯ is equal to 𝑑 into this function to find an expression for the right-hand side of this equation. First, if we substitute π‘₯ is equal to 𝑑 plus β„Ž into the function 𝑓 of π‘₯, we get π‘Ž times 𝑑 plus β„Ž all squared plus 𝑏 multiplied by 𝑑 plus β„Ž. Then, we need to subtract 𝑓 evaluated at 𝑑, which we can find by substituting π‘₯ is equal to 𝑑 into 𝑓 of π‘₯. We subtract π‘Žπ‘‘ squared plus 𝑏𝑑. This then gives us an equation in terms of π‘Ž, β„Ž, 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 β„Ž multiplied by 𝑑 plus β„Ž. We get 𝑑 squared plus two π‘‘β„Ž plus β„Ž squared. And we need to multiply this expression by π‘Ž. We can also distribute the negative over the two terms at the end of this expression. This gives us that π‘Žβ„Ž squared plus π‘β„Ž is equal to π‘Ž times 𝑑 squared plus two π‘‘β„Ž plus β„Ž squared plus 𝑏 times 𝑑 plus β„Ž minus π‘Žπ‘‘ squared minus 𝑏𝑑.

We can simplify the right-hand side further by distributing the factor of π‘Ž and the factor of 𝑏 over the parentheses. Doing this gives us the following equation, which we can simplify. We have π‘Žπ‘‘ squared minus π‘Žπ‘‘ 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 π‘Žβ„Ž squared on both the left- and right-hand side of the equation and 𝑏 times β„Ž on both the left- and right-hand side of the equation. So we can subtract π‘Žβ„Ž squared from both sides of the equation and 𝑏 plus β„Ž from both sides of the equation. This gives us that zero is equal to two times π‘Ž times 𝑑 multiplied by β„Ž.

So, for this equation to hold true, one of our unknowns must be equal to zero. It’s worth noting this can’t be β„Ž because β„Ž is a variable. It can take any value for our variation function. And π‘Ž 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 β„Ž would be for 𝑑 to be equal to zero. Therefore, we were able to show if the variation function of 𝑓 of π‘₯ is equal to π‘Žπ‘₯ squared plus 𝑏π‘₯ at π‘₯ is equal to 𝑑 is 𝑉 of β„Ž is equal to π‘Žβ„Ž squared plus π‘β„Ž, then the value of 𝑑 must be zero.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.