The \(x\)-axis is the line \(y = 0\), so look for the two places where the curve passes through it.

They are at \(x = 1\) and \(x = 5\).

Notice they are the same distance either side of the turning point at \(x = 3\) — \(2\) each way.

The \(y\)-axis is the line \(x = 0\), so read the height of the curve where it passes the upright axis.

The curve crosses at \((0, 5)\).

You can check it without the graph: putting \(x = 0\) into \(y = x^2 - 6x + 5\) kills both \(x\) terms and leaves \(y = 5\).

Substitute \(x = -4\) into the equation, working out the square before anything else:

\(y = (-4)^2 + 2(-4) = 8\)

Keep the brackets round the value you put in. Without them a negative \(x\) loses its sign in the squaring, which is the slip this line is written to stop.

\(y = (1)^2 + 2(1) = 3\)

Check it against the row. The \(y\) values fall to the lowest one and then climb back, and they climb back through the same numbers they came down.

Read along the completed \(y\) row for the smallest value. It is \(-1\), in the column headed \(x = -1\).

So the lowest point is \((-1, -1)\).

The table is symmetric about that column — the value one step to the left of it matches the value one step to the right — which is another way of spotting it.

The turning point is the lowest point of the curve — where it stops falling and starts rising.

From that point go straight up to the \(x\)-axis to read its \(x\)-coordinate, and straight across to the \(y\)-axis to read its \(y\)-coordinate.

The curve turns at \((1, -1)\).

A parabola is symmetric about the vertical line through its turning point. That turning point is at \(x = 1\), so the line of symmetry is

\(x = 1\)

It is an upright line, so its equation fixes \(x\), not \(y\). Writing \(y = 1\) is the usual slip.

Find \(3\) on the \(x\)-axis and go straight down until you meet the curve, then read across to the \(y\)-axis.

\(y = -8\)

Start on the \(y\)-axis at \(-5\) and go across. A horizontal line at that height meets the curve twice, once on each arm.

Read straight down from each meeting point to the \(x\)-axis:

\(x = 2\) and \(x = 6\)

Both answers matter. A curve is not a straight line, so one value of \(y\) usually comes from two different values of \(x\).

Each column of the table is one point: the \(x\) value across, the \(y\) value up.

Plot \((-2, 5)\), \((-1, 0)\), \((0, -3)\), \((1, -4)\), \((2, -3)\), \((3, 0)\) and \((4, 5)\).

Join them with one smooth curve, not with straight lines. The curve is symmetrical about \(x = 1\) and turns at its lowest point, \((1, -4)\).

Look at the steps from one \(y\) to the next. For a table like this one, centred on the lowest value, they should go

\(-5,\ -3,\ -1,\ +1,\ +3,\ +5\)

Two steps in a row disagree with that, and the entry they share is the one at fault: \(x = 1\).

The other check is symmetry. The value at \(x = 1\) should match the value at \(x = 5\), which is the same distance from the lowest point on the other side. It does not.

\(y = (1)^2 - 6(1) = -5\)

The table shows \(-4\), so it is out by \(1\).

The curve gives the value of \(x^2 + 6x + 8\) at every \(x\), and the flat line gives \(3\) at every \(x\). The two are equal exactly where the graphs cross.

They cross twice. Reading straight down from each crossing to the \(x\)-axis gives

\(x = -5\) and \(x = -1\)

A quadratic equation usually has two solutions, and here you can see why: a horizontal line cuts a parabola on both arms.

This curve rises to a peak and then falls, so its turning point is a maximum rather than a minimum.

Go straight down from the peak to the \(x\)-axis to read its \(x\)-coordinate, then straight across to the \(y\)-axis for its \(y\)-coordinate.

The peak is at \((1, 1)\).

The curve cuts the \(x\)-axis on the way up and again on the way down, at

\(x = 0\) and \(x = 2\)

They sit \(1\) either side of \(x = 1\), the line of symmetry through the peak.

The curve is on the \(x\)-axis when \(y = 0\), and a product of two brackets is zero only when one of the brackets is zero.

\(x + 3 = 0\) gives \(x = -3\)

\(x - 5 = 0\) gives \(x = 5\)

You do not have to multiply the brackets out — written this way, the equation hands you the crossings.

A parabola is symmetric, so its line of symmetry is exactly halfway between the two crossings.

Halfway between \(-3\) and \(5\) is

\(x = \frac{ -3 + (5) }{2} = 1\)

The lowest point of the curve is on the line of symmetry, so put \(x = 1\) into the equation:

\(y = (1 + 3)(1 - 5) = (4)(-4) = -16\)

The minimum value of \(y\) is \(-16\).

The two brackets are \(4\) and \(-4\) because \(1\) is \(4\) away from each crossing — one above, one below. That is why the minimum of a parabola with its roots \(8\) apart is always \(-(4)^2 = -16\).

A parabola is symmetric about its turning point, so the \(y\) row must read the same on either side of it.

The lowest value in the table, \(-13\), is printed twice: at \(x = 2\) and at \(x = 3\). Neither of those is the turning point, which is halfway between them.

\(x = \frac{ (2) + (3) }{2} = 2.5\)

The minimum is at the turning point, \(x = 2.5\). Substitute it:

\(y = 2(2.5)^2 - 10(2.5) - 1\)

\(y = 12.5 - 25 - 1\)

\(y = -13.5\)

The turning point lies between two columns, so the table never shows the minimum itself.