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 5 - 2017 - Paper 1

To sketch the graph of the function, we recognize that it is a parabola opening upwards, as the leading coefficient of the quadratic term is positive.

1. Identify the Vertex (Q): From the completed square form, the vertex (which is also the minimum point) is at (4, 3).

2. Y-Intercept (P): To find where the curve crosses the y-axis, set x = 0:

   • $f(0) = (0 - 4)^2 + 3 = 16 + 3 = 19$
   • So, point P is at (0, 19).

3. Plot Points: Mark points P (0, 19) and Q (4, 3) on the graph and sketch the parabola through these points. Ensure the curve is U-shaped and shows the correct orientation.

To find the distance between points P (0, 19) and Q (4, 3), we apply the distance formula:

$d = ext{PQ} = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Where:

• Point P = (0, 19) and Point Q = (4, 3)

Substituting the coordinates:

$d = \\sqrt{(4 - 0)^2 + (3 - 19)^2} = \\sqrt{4^2 + (-16)^2}$

Calculating further:

$d = \\sqrt{16 + 256} = \\sqrt{272}$

Upon simplifying, we get:

$d = \\sqrt{16 imes 17} = 4\sqrt{17}$

Thus, the distance PQ is expressed as a simplified surd: $4\sqrt{17}$.