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

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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

the gradient of a curve.

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.

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