Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$. The curve C meets $l$ at the points $(-2, 13)$ and $(0, 25)$ as shown. The shaded region $R$ is bounded by C and $l$ as shown in Figure 1. Given that - $f(x)$ is a quadratic function in $x$ - $(-2, 13)$ is the minimum turning point of $y = f(x)$ use inequalities to define R.

A-Level Edexcel Maths Pure Modelling with Exponentials & Logarithms: Figure 1 shows a sketch of a cur

Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$ - Edexcel - A-Level Maths Pure - Question 8 - 2020 - Paper 1

Question 8

Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$. The curve C meets $l$ at the points $(-2, 13)$ and $(0, 25)$ as shown. Th... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$ - Edexcel - A-Level Maths Pure - Question 8 - 2020 - Paper 1

Step 1

Finding the equation of the line $l$

96%

114 rated

Answer

To find the equation of the line $l$ , we need its slope and y-intercept.
The slope $m$ can be calculated using the two points $(-2, 13)$ and $(0, 25)$ :

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{25 - 13}{0 - (-2)} = \frac{12}{2} = 6.$

Using point-slope form $y - y_1 = m(x - x_1)$ , we can substitute one of the points, for example $(0, 25)$ :

$y - 25 = 6(x - 0)$
which simplifies to
$y = 6x + 25.$

Step 2

Finding the equation of the curve $C$

99%

104 rated

Answer

Since $f(x)$ is a quadratic function with a minimum turning point at $(-2, 13)$ , we can express it in vertex form:

$f(x) = a(x + 2)^2 + 13.$
To determine the value of $a$ , we use another point the curve passes through, $(0, 25)$ :

Substituting $(0, 25)$ into the equation:

$25 = a(0 + 2)^2 + 13$
$25 = 4a + 13$

$$ a = 3.$$ The equation of the curve now is: $$f(x) = 3(x + 2)^2 + 13.$$

Step 3

Define the region $R$

96%

101 rated

Answer

The region $R$ is bounded by the line $l$ and the curve $C$ . For the inequality:

$f(x) \leq l$ Substituting the expressions we found:

$3(x + 2)^2 + 13 \leq 6x + 25.$
Rearranging gives:

\Rightarrow 3(x + 2)^2 - 6x - 12 \leq 0.$ Further simplification will help solve the inequality to find the x-values of region $R$.

Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$ - Edexcel - A-Level Maths Pure - Question 8 - 2020 - Paper 1

Worked Solution & Example Answer:Figure 1 shows a sketch of a curve C with equation $y = f(x)$ and a straight line $l$ - Edexcel - A-Level Maths Pure - Question 8 - 2020 - Paper 1

Finding the equation of the line $l$

Finding the equation of the curve $C$

Define the region $R$

Join the A-Level students using SimpleStudy...