Question Video: Finding the Value of the Variation Function of a Quadratic Function | Nagwa Question Video: Finding the Value of the Variation Function of a Quadratic Function | Nagwa

# Question Video: Finding the Value of the Variation Function of a Quadratic Function Mathematics • Second Year of Secondary School

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If π is the variation function for π(π₯) = π₯Β² β 4π₯ + 2, what is π(β0.2) when π₯ = 8?

02:45

### Video Transcript

If π is the variation function for π of π₯ is equal to π₯ squared minus four π₯ plus two, what is π of negative 0.2 when π₯ is equal to eight?

Weβre given a quadratic function π of π₯ is π₯ squared minus four π₯ plus two and asked to find the value of the variation function if π₯ changes from π₯ is equal to eight by an amount of negative 0.2. The first thing we need to do then is to find the variation function π of β. We can do this using the definition for a function π of π₯ where its variation function at π₯ is equal to π is given by π of β is π of π plus β minus π of π. And thatβs where β is the change in π₯. In our case, our function π of π₯ is π₯ squared minus four π₯ plus two. Our given value of π₯ is eight so that π is equal to eight so that our variation function π of β is π of eight plus β minus π of eight.

There are two ways we could approach this problem. We know that our change in π₯ is negative 0.2, and this means that β is negative 0.2. And we could substitute this directly into our equation for π of β. Alternatively, we could first find π of β, the variation function in terms of β, and then substitute β is negative 0.2. Weβre going to use the second method to find the variation function π of β. And to do this, weβre first going to substitute π₯ is equal to eight plus β into our function π of π₯. And this gives us eight plus β squared minus four times π plus β plus two. Distributing our parentheses, this gives us 64 plus 16β plus β squared minus 32 minus four β plus two. And collecting like terms, this gives us eight squared plus 12β plus 34.

Now, to find π of eight, we substitute π₯ equal to eight into our function π of π₯, which gives us eight squared minus four times eight plus two. And this evaluates to 34. Now, with these two results into our function π of β, we have eight squared plus 12β plus 34 minus 34, that is, β squared plus 12β. And this is our variation function π of β for π of π₯ at π₯ is equal to eight. Now, to find π of negative 0.2, we substitute negative 0.2 in place of β. This gives us negative 0.2 squared plus 12 times negative 0.2. That is 0.04 minus 2.4, which is negative 2.36. π of negative 0.2 for π of π₯ is equal to π₯ squared minus four π₯ plus two when π₯ is equal to eight is therefore negative 2.36.

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