7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$. 7 (b) The function $f(x)$ is defined by $f(x) = x^3 + 3px^2 + q$ where $p$ and $q$ are constants and $p > 0$. 7 (b) (i) Show that there is a turning point where the curve crosses the $y$-axis.

A-Level AQA Maths Mechanics Variable Acceleration - 1D: 7 (a) Sketch the graph of any cu

7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ - AQA - A-Level Maths Mechanics - Question 7 - 2019 - Paper 2

Question 7

7-(a)-Sketch-the-graph-of-any-cubic-function-that-has-both-three-distinct-real-roots-and-a-positive-coefficient-of-$x^3$-AQA-A-Level Maths Mechanics-Question 7-2019-Paper 2.png

7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$. 7 (b) The function $f(x)$ is defined by ... show full transcript

Worked Solution & Example Answer:7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ - AQA - A-Level Maths Mechanics - Question 7 - 2019 - Paper 2

Step 1

Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$

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Answer

To sketch the graph of a cubic function with three distinct real roots and a positive leading coefficient, we note that the graph will have the following characteristics:

Positive Leading Coefficient: The end behavior of the cubic function will rise to the right and rise to the left.
Distinct Real Roots: The graph will cross the x-axis at three points, indicating three distinct real roots, denoted as $r_1$ , $r_2$ , and $r_3$ . The graph will change direction at each root.
Sketch Representation:
- Start with three x-intercepts ( $r_1$ , $r_2$ , $r_3$ ) marked on the x-axis.
- The graph typically will dive down from the left, cross the x-axis at $r_1$ , rise upward, cross again at $r_2$ , and finally fall, crossing at $r_3$ before rising off to the right.

This results in a graph that smoothly curves up and down, representing the positive coefficient and the three distinct real roots.

Step 2

Show that there is a turning point where the curve crosses the $y$-axis.

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Answer

To show that there is a turning point where the curve crosses the $y$ -axis, we start with the given function:

$f(x) = x^3 + 3px^2 + q$

Differentiate the Function: $f'(x) = 3x^2 + 6px$
We need to find the turning points where $f'(x) = 0$ .
Set Derivative to Zero: $3x^2 + 6px = 0$
Factor out common terms: $3x(x + 2p) = 0$ This gives us the solutions:
- $x = 0$
- $x = -2p$
Analyzing the Turning Points:
- Since $p > 0$ , it follows that $-2p < 0$ .
- We have established turning points at $x = 0$ and $x = -2p$ .
Evaluate the Function at $x = 0$ : $f(0) = q$
This indicates that when the curve crosses the $y$ -axis (at $x = 0$ ), the value of the function is $f(0) = q$ .
Determine the Nature of the Turning Point at $x = 0$ :
- At $x = 0$ , if $f'(0) = 0$ and $f(0) = q$ , then this point can act as a minimum or maximum.

Since we can see that there is a turning point at $x=0$ where the curve crosses the $y$ -axis, we have shown that such a point exists.

7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ - AQA - A-Level Maths Mechanics - Question 7 - 2019 - Paper 2

Worked Solution & Example Answer:7 (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$ - AQA - A-Level Maths Mechanics - Question 7 - 2019 - Paper 2

Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x^3$

Show that there is a turning point where the curve crosses the $y$-axis.

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