f(x) = 2x² + 4x + 9 (a) Write f(x) in the form α(x + b)² + c, where a, b and c are integers to be found - Edexcel - A-Level Maths Pure - Question 7 - 2019 - Paper 2

To sketch the curve, we need to identify key features:

1. Y-intercept: Set x = 0:

   $f(0) = 2(0)² + 4(0) + 9 = 9$

   Thus, the y-intercept is (0, 9).

2. Turning Point: The minimum value occurs at the vertex:

   $x = -b/(2a) = -2/4 = -0.5$

   Plugging this back into f gives:

   $f(-0.5) = 2(-0.5)² + 4(-0.5) + 9 = -1.5 + 9 = 7$

   The turning point is at (-1, 7).

3. Sketch: The curve passes upward through the y-axis at (0, 9) and has a minimum turning point at (-1, 7), indicative of a U-shaped graph.

The transformation can be derived as follows:

1. Starting with the function f(x):

   $f(x) = 2(x + 1)² + 7$

2. The equation for g(x) is given by g(x) = 2(x - 2)² + 4x - 3.
   This indicates a combination of translations and stretches.

3. The transformation can be described as follows:

   • Horizontal Translation: Shift the graph 2 units to the right.
   • Vertical Stretch: The factor of 2 indicates that the graph is vertically stretched by a factor of 2.
   • Vertical Translation: Upward shift to adjust for the change in minimum and maximum values.

To find the range of the function h(x) = \frac{21}{2x² + 4x + 9}, we first analyze the denominator 2x² + 4x + 9:

1. The quadratic opens upwards (since the coefficient of x² is positive) and has a minimum value at:

   $x = -\frac{b}{2a} = -\frac{4}{2(2)} = -0.5$

2. Plugging this value into the quadratic gives:

   $f(-0.5) = 2(-0.5)² + 4(-0.5) + 9 = 7$

3. Thus, the minimum value of the denominator is 7, and

   $h(x) < \frac{21}{7} = 3$

4. Therefore, the range of h(x) is:

   $0 < h(x) < 3$