This material is an extract from our National 5 Mathematics: Curriculum Breakdown course led by instructor Andrew Eadie. Enrol in the full course now and gain access to over 100 detailed topic breakdowns, 48 video tutorials (20 hours) and 39 quizzes spanning the entire curriculum. Just like with straight lines, the topic of quadratics (or quadratic functions, or parabolas – all these terms mean the same thing), ultimately boils down to a single equation – the General Equation of a Quadratic Function:

The General Equation of a Quadratic Function is:

#### \boldsymbol{y=ax^{2}+bx+c}

Where:

\boldsymbol{x} and \boldsymbol{y} are variables; and
\boldsymbol{a}, \boldsymbol{b} and \boldsymbol{c} are constants (often called “coefficients”) which change the curvature and position of the quadratic

Unlike with straight lines, where the constants m and c (the gradient and y-intercept respectively) had names, the coefficients a, b and c don’t have separate names (unhelpfully…)

x and y are simply coordinates on typical xy axes, and given that they are variables, x and y will change depending on the position you’re looking at on a quadratic graph. However, for any specific quadratic function, the coefficients \boldsymbol{a}, \boldsymbol{b} and \boldsymbol{c} are fixed.

As you no doubt already know, quadratics are not straight lines – they are curved lines with a line of symmetry through their turning point. As an example, the quadratic y=x^{2}-6x+8 is shown below: But why are they curved? What is it about quadratics that makes them curved instead of straight?

It’s all because of the x^{2} term. In straight lines, which can all be written in the form of the General Equation y=mx+c, there is no powered x term. Once you start introducing x terms with powers such as x^{2}, x^{3}, x^{4} and so on, the behaviour of graphs changes drastically and they become much more complicated than simple straight lines.

Let’s compare the simplest straight line, y=x (which we met in the previous chapter), with the simplest quadratic, y=x^{2}, to understand how quadratics end up being curved instead of straight.

We can sketch both functions using a very basic coordinate table method. We can create coordinate tables by substituting different values of x into each function to calculate the corresponding values of y and then plot the results as coordinates in the form (x, y). Here it is useful to look at both positive and negative values of x for both functions: 1) Choose some x values: -3, -2, -1, 0, 1, 2, 3

2) Substitute these values into the equation to calculate corresponding y values:

\begin{aligned}y&=x \\[12pt]y&=(-3)=-3 \\[12pt]y&=(-2)=-2 \\[12pt]y&=(-1)=-1 \\[12pt]y&=(0)=0 \\[12pt]y&=(1)=1 \\[12pt]y&=(2)=2 \\[12pt]y&=(3)=3\end{aligned}

3) Summarise results in a coordinate table: 4) Plot the points on xy coordinate axes: There should be nothing surprising about this graph at this stage – it is a simple straight line with all equal coordinates, a gradient of 1 and a y-intercept of (0, 0).

Notice that because the gradient is 1, every time x increases or decreases by 1 then y also increases or decreases by 1: Repeating this process with the simplest quadratic function y=x^{2}, we have the following: 1) Choose some x values: -3, -2, -1, 0, 1, 2, 3

2) Substitute these values into the equation to calculate corresponding y values:

\begin{aligned}y&=x^{2} \\[12pt]y&=(-3)^{2}=9 \\[12pt]y&=(-2)^{2}=4 \\[12pt]y&=(-1)^{2}=1 \\[12pt]y&=(0)^{2}=0 \\[12pt]y&=(1)^{2}=1 \\[12pt]y&=(2)^{2}=4 \\[12pt]y&=(3)^{2}=9\end{aligned}

3) Summarise results in a coordinate table: 4) Plot the points on xy coordinate axes: This graph shows some very different behaviour, and it is quite clearly non-linear. Since any number squared is always positive, there are no negative y values even when x is negative, which results in a line of symmetry running through the turning point. In this case, the turning point is the coordinate (0, 0) and the line of symmetry is the y-axis itself.

If someone asked you what the gradient of this quadratic is, what would you say? That’s sort of a confusing question isn’t it? Because, unlike straight lines which have an obvious slope which does not change, a quadratic is sloped differently at different points. At some points the graph is very steep, whilst at others the graph flattens out. Because of this, when x increases or decreases by 1, then y does not always increase or decrease by an equivalent amount.

This is what causes the graph to become curved. Looking again at the coordinate table for y=x^{2} and starting from the x=0 position, we can see that for every jump of 1 in x, y does not always move by the same amount:  What this exercise shows is that including an x^{2} term in a function causes it to become a symmetric curve. The \boldsymbol{x^{2}} term is what gives the quadratic its characteristic symmetrically curved shape. If there’s no \boldsymbol{x^{2}} term, then a function cannot be a quadratic.

The quadratic we examined above, y=x^{2}, is the simplest quadratic function. Although it may not look like it at first, this quadratic is in fact in the form of the General Equation y=ax^{2}+bx+c. If we write our quadratic y=x^{2} in a more expanded form, this becomes clearer:

\begin{aligned}y&=x^{2} \\[12pt]y&=1x^{2}+0x+0\end{aligned}

Do the changes I have made actually do anything? No, they don’t – these two lines say exactly the same thing! Explicitly showing a coefficient of 1 before the x^{2} term doesn’t do anything since multiplying by 1 has no effect, and any term with a 0 attached to it will just disappear since multiplying by 0 always gives 0. However, comparing this to the General Equation it is now clearer that our original quadratic is in fact in standard form:

\begin{aligned}y&=x^{2} \\[12pt]y&=1x^{2}+0x+0 \\[12pt]y&=ax^{2}+bx+c\end{aligned}

The simplest quadratic, y=x^{2}, therefore has coefficients a=1, b=0 and c=0.

If we were to change the \boldsymbol{a}, \boldsymbol{b} and \boldsymbol{c} coefficients of \boldsymbol{y=x^{2}} or any other quadratic function, the function’s curvature and position will be affected.

Compare the graph of the simplest quadratic, y=x^{2}, which has coefficients a=1, b=0 and c=0, to the graph of another random quadratic, say y=4x^{2}-12x+10, which has coefficients a=4, b=-8 and c=6, and you can see there are clear differences in the curvature (or steepness) and position of the two graphs: All quadratics have 4 key features:

1) Nature
2) \boldsymbol{y}-intercept
3) \boldsymbol{x}-intercepts (or “roots”); and
4) Turning point

These 4 features are highlighted on the quadratic y=x^{2}-4x+3 shown below: In the following topics, we will explore how the a, b and c coefficients of a quadratic impact these 4 features.

### Key Outcomes

Quadratics are not straight lines – they are curved lines with a line of symmetry through their turning point.

Any quadratic function can be expressed in the standard form of the General Equation:

#### \boldsymbol{y=ax^{2}+bx+c}

Where:

x and y are variables; and
a, b and c are constants (often called “coefficients”) which change the curvature and position of the quadratic

The x^{2} term is what gives the quadratic its characteristic symmetrically curved shape. If there’s no x^{2} term, the function is not a quadratic.

For any specific quadratic function, the coefficients a, b and c are fixed.

To understand any quadratic in full, we look at its 4 key features:

1) Nature
2) y-intercept
3) x-intercepts (or “roots”); and
4) Turning point

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