Quadratic and Cubic Graphs

1. What does the Equation of a Curve Actually Mean?

The equation of a curve, whether it be a quadratic, cubic, or any other type, is a way of expressing the relationship between the $x-$ coordinates and the $y-$ coordinates that lie on that curve.

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Example: For the equation:

$y=x^2+3x−9$

This says that the relationship between all the x_x_-coordinates and all the $y-$ coordinates is: "get your $x-$ coordinate, square it, add on three lots of your $x-$ coordinate, subtract $9$ , and you get your $y-$ coordinate."

If a pair of coordinates like ( $2,1$ ) has this relationship, then it's on the curve.
If it doesn't, such as $(5,4)$ , then it does not lie on the curve. What you end up with is just a curve that goes through all the coordinates which share that relationship.

2. Drawing Curves from their Equation

The method of drawing curves is very similar to drawing straight lines, with just a few more points needed to capture the shape.

Step-by-Step Process:

Choose a Sensible Value of $x$ :

Select values that are small enough to fit on the paper and easy to work out.

Substitute $x$ into the Equation:

Substitute each chosen $x-$ value into the equation to calculate the corresponding $y-$ value.

Repeat for Multiple Points:

Do this enough times (typically $5-7$ points) to see the shape of the curve.

Plot and Join the Points:

Plot the points on the graph and join them up with a smooth curve (no sharp, pointy bits).

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Crucial Tip:

You are more likely to get the shape of the curve right if you have a good knowledge of what shapes different equations make. Review graph shapes before you start plotting.

Common Mistake:

Be careful with negative numbers—mistakes often happen here. Whether you are doing this on a calculator or in your head, double-check your signs!

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Final Tip:

It's often easier to pick $x=0$ as one of your points since it's easy to calculate $y$ for $x=0$ .

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Worked Example:

Given Equation:

y=x^2+3x−9

Step-by-Step Solution:

Choose $x-$ values:

For instance, choose $x=−2,−1,0,1,2$ .

Substitute and Calculate $y-$ values:

For $x=−2$ :

y=(−2)^2+3(−2)−9=4−6−9=−11

Continue for other $x-$ values.

Create a Table:

\begin{array}{c|c}x & y \\\hline-2 & -11 \\-1 & -7 \\0 & -9 \\1 & -5 \\2 & 1 \\\end{array}

Plot Points and Draw the Curve:

Plot these points on a graph and connect them smoothly to show the shape of the curve.

3. Substituting Numbers in Your Head

When you are asked to draw a curve on a non-calculator paper, it is crucial to be very careful with your calculations. Here's a step-by-step guide to help you substitute values into equations accurately in your head.

Things to Remember:

Order of Operations: Always remember BODMAS (Brackets, Orders (i.e., powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right)).
Rules of Negative Numbers: Be particularly cautious when dealing with negative numbers, as it's easy to make mistakes here.

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Worked Example: Let's substitute $x=−2$ into the quadratic equation $y=x^2−4x+2$ . Here's how you would work it out in your head:

Squared Term:

Calculate $(−2)^2$ , which equals $4$ .
Remember, squaring a negative number gives you a positive result.

Multiply by $x$ :

Next, you deal with the $−4x$ term. Since $x=−2$ , you calculate $−4×(−2)$ .
This equals $+8$ (multiplying two negatives gives a positive).

Combine All Terms:

Now, substitute into the equation: $y=4−8+2.$
Simplify step by step:
$4−8=−4$
$−4+2=−2$ So, the final result is $y=−2$ .

Coordinate Pair:

The point you need to plot on your graph is $(−2,−2)$ .

4. Substituting Numbers Using a Calculator

While having a calculator makes doing tricky calculations much easier, it also means you are likely to get much more difficult numbers to work with. If you're not careful, calculators can lead to mistakes as well. Here's how to avoid those mistakes and ensure your calculations are correct.

Things to Remember:

Put Negative Numbers in Brackets: Always use brackets for negative numbers to avoid any confusion or errors. For example, if you're inputting $−4$ , type it as $(−4)$ in your calculator.
Double-Check Your Calculations: Always do each calculation twice to make sure you didn't press a wrong button. This simple step can save you from making easy-to-avoid mistakes.

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Worked Example: Let's substitute $x=−4$ into the equation $y=x^3+2x^2−6x+2$ . Here's how you should input this into your calculator:

First Term $x^3$ :

Input: $(−4)^3$
Result: −64

Second Term $2x^2$ :

Input: $2×(−4)^2$
Result: $32$

Third Term $−6x$ :

Input: $−6×(−4)$
Result: $24$

Constant Term $+2$ :

This is just $+2$ .

Putting it all together:

Combine all the results: $y=−64+32+24+2$

This simplifies to: $y=−6$

So, by following the correct steps, the value of $y$ when $x=−4$ is −6.

5. Using Curves to Solve Equations

Once you've taken the time to draw a curve, you can use it to solve an equation. Here's how you do it:

Method:

Re-arrange the Equation:

Ensure that all the letters are on the left-hand side, and there is either a number or zero on the right-hand side.

Draw the Graph:

Plot the graph of the left-hand side of the equation.

Draw a Horizontal Line:

On your graph, draw a horizontal line at the level of the number on the right-hand side of the equation.

Mark Intersection Points:

Identify and mark the points where this horizontal line crosses your curve.

Find the Solutions:

The $x-$ coordinates of these points are the solutions to the equation. Note: If there is a zero on the right-hand side of the equation, you are looking for the points where the curve crosses the $x-$ axis.

6. Putting it all Together

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Example 1: Solving Quadratic Equations Using Graphs In this example, we will learn how to solve quadratic equations by using their graphs.

Problem: Solve the equation $x^2−3x−4=0$ using a graph.

Steps:

Plot the Quadratic Function:

The equation of the quadratic function is $y=x^2−3x−4$ .
Use a table of values to plot the graph. For example:

Plot these points on a graph and draw a smooth curve through them.

Identify the $x-$ Intercepts:

The solutions to the equation $x^2−3x−4=0$ are the points where the graph crosses the $x$ -axis.
From the graph, you can see that the curve crosses the $x$ -axis at $x=1$ and $x=4.$ Write the Solutions:
The solutions to the equation $x^2−3x−4=0$ are $x=1$ and $x=4$

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Example 2: Solving Cubic Equations Using Graphs In this example, we will learn how to solve cubic equations by using their graphs.

Problem: Solve the equation $x^3−8x+5=0$ using a graph.

Steps:

Plot the Cubic Function:

The equation of the cubic function is $y=x^3−8x+5$ .
Use a table of values to plot the graph. For example:

Plot these points on a graph and draw a smooth curve through them.

Identify the $x$ -Intercepts:

The solutions to the equation $x^3−8x+5=0$ are the points where the graph crosses the $x-$ axis.
From the graph, you can see that the curve crosses the $x$ -axis at approximately $x=−3.1$ , $x=0.7$ , and $x=2.5$ . Write the Solutions:
The solutions to the equation $x^3−8x+5=0$ are approximately $x=−3.1$ , $x=0.7$ , and $x=2.5$ . image.png

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Example 3: Solving Quadratic Equations Using Graphs In this example, we will solve a quadratic equation using its graph and a horizontal line.

Problem: Solve the equation $2x^2−5x=4$ using a graph.

Steps:

Plot the Quadratic Function:

The equation of the quadratic function is $y=2x^2−5x.$
Use a table of values to plot the graph. For example:

Plot these points on a graph and draw a smooth curve through them.

Draw the Line $y=4$ :

Draw a horizontal line on the graph where $y=4$ .
The points where this line intersects the curve of the quadratic function represent the solutions to the equation $2x^2−5x=4$ . Identify the $x-$ Intercepts:
From the graph, you can see that the curve crosses the horizontal line $y=4$ at approximately $x=−0.7$ and $x=3.2$ . Write the Solutions:
The solutions to the equation $2x^2−5x=4$ are approximately $x=−0.7$ and $x=3.2$ .

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Quadratic and Cubic Graphs Simplified Revision Notes for GCSE OCR Maths

Quadratic and Cubic Graphs

1. What does the Equation of a Curve Actually Mean?

2. Drawing Curves from their Equation

Worked Example:

3. Substituting Numbers in Your Head

4. Substituting Numbers Using a Calculator

5. Using Curves to Solve Equations

6. Putting it all Together

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