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Given that f(x) = x² - 4x + 5 x ∈ ℝ a) express f(x) in the form (x + a)² + b where a and b are integers to be found - Edexcel - A-Level Maths Pure - Question 4 - 2021 - Paper 1
Question 4
Given that f(x) = x² - 4x + 5 x ∈ ℝ a) express f(x) in the form (x + a)² + b where a and b are integers to be found. The curve with equation y = f(x) • meets t... show full transcript
Worked Solution & Example Answer:Given that f(x) = x² - 4x + 5 x ∈ ℝ a) express f(x) in the form (x + a)² + b where a and b are integers to be found - Edexcel - A-Level Maths Pure - Question 4 - 2021 - Paper 1
Step 1
express f(x) in the form (x + a)² + b
Answer
To express the function in the required form, we need to complete the square.
Starting with:
f(x) = x^2 - 4x + 5 $$ We can complete the square as follows: 1. Identify the coefficient of x, which is -4. 2. Take half of this coefficient and square it: $$\left(\frac{-4}{2}\right)^{2} = 4$$. 3. Rewrite the function: $$f(x) = (x^2 - 4x + 4) + 5 - 4$$ $$f(x) = (x - 2)^2 + 1$$ Thus, we have: $$f(x) = (x - 2)^2 + 1$$ where a = -2 and b = 1.Step 2
Step 3
the coordinates of Q
Answer
Since point Q is the minimum turning point, we can find the coordinates by recognizing that this occurs at the vertex of the parabola given by the completed square form. The vertex occurs at:
Substituting this value back into f(x) gives:
.
Thus, the coordinates of Q are (2, 1).