Quadratic Graphs

A quadratic has an x^2 term, and no higher power of x. Examples are y=x^2+2x-4 and y=x^2-5. The graph of a quadratic is a smooth U-shaped curve. It is never a straight line.

In the exam you are usually asked to complete a table of values, for 2 marks, and draw the graph, for 2 marks. Then you write down the turning point, for 1 mark, or use the graph to find estimates for the solutions of an equation, for 2 marks. Some questions give you the graph already drawn, and now and then you are asked to sketch a graph.

To complete a table of values, put each value of x into the equation, one column at a time. Work out each term on its own, then combine them.

Put a negative value of x in a bracket. Squaring a negative number gives a positive answer: (-3)^2=(-3)\times(-3)=9, not -9. On a calculator, type the bracket too, because -3^2 without a bracket gives -9

Don't fill in the table by carrying on a pattern, as you can for a straight line. The gaps between the y-values change, so work out every value from the equation.

To check whether a point is on the graph, put its coordinates into the equation. For example, (-3,5) is on y=x^2-4, because (-3)^2-4=9-4=5

Check the scale on each axis, then plot each point from the table with a small cross. Take extra care with points that have negative coordinates.

Join the points with one smooth curve, drawn freehand. Don't use a ruler, and don't join the points with straight lines. Go through every point, from the first value of x to the last.

The bottom of a U-shaped curve is always rounded. When the two lowest points are level, as here at (2,-2) and (3,-2), the curve dips a little below them, halfway between. Never draw a flat bottom.

If one point does not fit a smooth U-shape, work it out again before you draw. Never bend the curve to reach a wrong point.

The turning point is where the curve turns round. On a U-shape it is the lowest point. Here it is (3,-4). Write it as coordinates, with x first.

The curve crosses the y-axis where x=0. Putting x=0 into y=x^2-6x+5 leaves just the number at the end, so the y-intercept is (0,5)

The roots of the equation x^2+2x-4=0 are the values of x that make it true. On the graph of y=x^2+2x-4, they are where y=0, which is where the curve crosses the x-axis.

When a U-shaped curve dips below the x-axis, it crosses it twice, so there are two roots. Give the two values of x as numbers, not as coordinates.

Before you read a value, check the scale on each axis. Count the small squares from one numbered line to the next. If there are 5, each small square is 0.2. If there are 10, each small square is 0.1. The two axes can have different scales. Your answer is an estimate, so a small range of values either side is accepted.

When a question says “use the graph”, read the values from the graph. Don't try to solve the equation with algebra.

To find y for a given value of x, go straight up or down from the x-axis to the curve, then straight across to the y-axis.

An equation such as x^2+2x-4=2 asks where the graph of y=x^2+2x-4 is at a height of 2. Draw the horizontal line y=2 across the grid. Then, from each point where it crosses the curve, go straight down or up to the x-axis and read the value.

Any horizontal line above the turning point crosses a U-shaped curve twice, so there are two answers. The examiners say many students give only one of them.

When the equation has only x^2 and a number, you can work the values out. If x^2-1=24, then x^2=25, so x=5 or x=-5. Squaring a negative number gives a positive answer, so don't forget the negative value.

A sketch shows the shape of a graph and its important points. You don't plot points or use a grid, but the shape must be right and the important points must be labelled.

The graph of y=x^2 plus or minus a number is a U-shape with its turning point on the y-axis. For y=x^2+3, putting x=0 gives y=3, so the turning point is (0,3)

You may be asked to match equations to sketch graphs. An equation with an x^2 term, such as y=x^2+3x, gives a U-shape. An equation such as y=2x+1, with just x and numbers, gives a straight line.

• Squaring a negative number wrongly. Put it in a bracket: (-2)^2=4, not -4.
• Filling in the table from a pattern. Work out every y-value from the equation.
• Joining the points with straight lines. Draw one smooth curve, freehand, through every point.
• Drawing a flat bottom. Between two level lowest points, the curve dips a little below them.
• Writing the turning point the wrong way round. Write x first, then y, and keep any minus signs.
• Giving one solution, or giving coordinates. Read both x-values where the curve crosses the x-axis or your line, and write them as numbers.