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f(x) = x² - 8x + 19 (a) Express f(x) in the form (x + a)² + b, where a and b are constants - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 1
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f(x) = x² - 8x + 19 (a) Express f(x) in the form (x + a)² + b, where a and b are constants. The curve C with equation y = f(x) crosses the y-axis at the point P a... show full transcript
Worked Solution & Example Answer:f(x) = x² - 8x + 19 (a) Express f(x) in the form (x + a)² + b, where a and b are constants - Edexcel - A-Level Maths Pure - Question 7 - 2017 - Paper 1
Step 1
Express f(x) in the form (x + a)² + b
Answer
To express the quadratic function in the form (x + a)² + b, we can complete the square.
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Start with the given function:
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To complete the square for the expression , take half of the coefficient of , which is , and square it to get .
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Rewrite the function:
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This results in:
Thus, we have and .
Step 2
Sketch the graph of C showing the coordinates of point P and the coordinates of point Q
Answer
The curve C is a parabola that opens upwards, with its vertex representing the minimum point Q.
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The coordinates of point P, where the curve crosses the y-axis, can be found by evaluating : Therefore, P(0, 19).
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The coordinates of point Q, which is the vertex, are (4, 3).
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When sketching the graph, ensure to mark:
- Point P at (0, 19)
- Point Q at (4, 3)
The graph should be a U-shaped curve that is symmetrical about the vertical line x = 4.
Step 3
Find the distance PQ, writing your answer as a simplified surd
Answer
To find the distance between points P(0, 19) and Q(4, 3), we can use the distance formula:
d = ext{PQ} = rac{ ext{distance between } P ext{ and } Q}{ ext{distance formula}}
d = ext{PQ} = rac{(x_2 - x_1)² + (y_2 - y_1)²}{ ext{where } (x_1, y_1) = (0, 19) ext{ and } (x_2, y_2) = (4, 3)}
- Substitute the coordinates into the formula: d = ext{PQ} = rac{(4 - 0)² + (3 - 19)²}{ ext{which simplifies to}} d = ext{PQ} = rac{4² + (-16)²}{ ext{which gives us}} d = ext{PQ} = rac{16 + 256} = rac{272} d = PQ = rac{16 ext{√}17}{7}
Final distance, written as a simplified surd: d = rac{4 ext{√}17}{7}.