• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
• Level: GCSE
• Subject: Maths
• Word count: 1745

# To investigate the effect of the constants a, b and c on the graph y = ax2 + bx + c.

Extracts from this document...

Introduction

## Maths C/W

Preparatory:

AIM: To investigate the effect of the constants a, b and c on the

graph y = ax2 + bx + c

PLAN:

To begin with, I will fix constants a and b, and will plot several equations with varying c values. I will then make observations, and attempt to prove any theories I discover. I will find out what happens when c increases and decreases, making sure to include negative values of c. I will examine any movement changes when the equation is linear or quadratic. I will try to prove that the graph is symmetrical, and where the minimum point is. I will also try to prove the y intercept. I will then summarise my findings.

After that, I will fix constants b and c, and plot equations with varying a values. I will make observations on these graphs, and try to prove any theories I may discern. I will try to explain my findings using diagrams, and will draw up a conclusion. I will draw tangents to certain curves and make observations on the gradients. I will then use calculus to check my tangents, and to also find out the general formula for the gradient of y = ax2. I will make sure to include negative values of a.

I will then fix constants a and c, and focus on constant b.

Middle

To summarise, we can safely conclude that modifying the constant c moves the graph up or down by the quantity of c (i.e. each point of the parabola has been increase by the value of c). The graph y = x2 + c will always cross the y-axis at c. We can also see that modifying c alone does not affect the gradient of the parabola, nor does it stop it from being symmetrical – the y-axis always being the line of symmetry.

Constant A

Next, we will focus on varying the constant a. We will keep b = 0, and will fix c = 0. If we then start off with a = 1, then we will be given the basic parabola, y = x2. On the same graph, we will plot a as other varying figures, making sure to include minus numbers and fractions.

As we can see, changing a clearly changes the gradient of the curve, except at x = 0 when the gradient will always be 0.. Higher values of a give steeper gradients, fractions give shallower ones. But by how much is the graph affected? To answer this we must begin by measuring the gradient - we can do this one of two ways.

Tangents to y = ax2

By drawing tangents

to the graph at certain

points, we can measure

Hand drawn tangents

can be inaccurate, however,

Conclusion

Path of y = ax2 +bx

Before we continue, we must note that the path that the turning point moves along when constant b is modified, changes when we modify constant a. Above, we saw that when a is negative, our turning point moves in a different direction along a different path. We will now go into further detail on the turning point of y = ax2 +bx, looking at graphs where a is not only positive or negative, but also a value other than 1 or -1.

On the right is plotted the equation

y = 3x2 – 10x. Here we can see that the turning point does not follow the path y = -x2 as before, when we fixed a as 1, but instead, it now follows the path y = -3x2.

##### Changing constants A, B and C simultaneously

We will now look at what happens when we change all three constants at the same time, and we will predict what a few graphs involving all three variables would look like. First we will go through several proofs of more precise facts about turning points and x/y intercepts.

We will now go through a step-by-step plotting of a graph, emphasizing any changes each constant might have.

To begin with, we will try to plot the graph of y = 3x2 – 9x +  7. We will first plot the graph y = x2and then apply our theories about each constant in turn.

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Gradient Function essays

1. ## Curves and Gradients Investigation

- x4 (expand and h simplify) = 4x³h + 6x²h² + 4xh³ + h4 (cancel x4) h = 4x³ + 6x²h + 4xh² + h³ (cancel h) as h tends to 0 GF tends to 4x³ Results Summary From calculating the gradient functions of the curves: y = x², y

1 x 3 = 3 I have now worked out the gradient of when the tangent is at x=-1, the gradient is 3. I have basically used the method, which I have stated below: 3 * x2 The 'star' in the equation, basically represents multiply.

1. ## Investigate gradients of functions by considering tangents and also by considering chords of the ...

In order to avoid this, I am going to use another method to calculate the gradients: increment method. Increment method works like this: you need 4 numbers: x1, x2, y1, y2. If you want to get the gradient of the co-ordination (1, 1), namely (x1, y1)

2. ## The Gradient Function Investigation

There will be n of these terms as there are always n terms in the second bracket. as h tends to 0 GF tends to Ax(n-1). My Final Goal My final aim for this investigation was to find and prove a general rule for curves of the form y = Axn + Bxm + C (where C is a constant).

1. ## Maths Coursework - The Open Box Problem

Cut Out x Width 30-2x Length 10-2x Volume 5.1 19.8 19.8 1999.404 5.2 19.6 19.6 1997.632 5.3 19.4 19.4 1994.708 5.4 19.2 19.2 1990.656 5.5 19 19 1985.5 5.6 18.8 18.8 1979.264 5.7 18.6 18.6 1971.972 5.8 18.4 18.4 1963.648 5.9 18.2 18.2 1954.316 6 18 18 1944 After zooming

axn anx(n-1) I think anx(n-1) is the gradient function for all y=axn graphs. To test if this is right I will first use this function to work out the gradient of certain points on the graphs. I am going to use the graphs - x5, 2x5, 3x5, 4x5.