Why is the turning point of y equals (x minus 7) squared plus 8 at x equals 7 and not x equals negative 7?

Because the turning point is where the squared bracket equals zero. Setting (x minus 7) equal to zero gives x equals 7. The x-coordinate is the value that makes the bracket zero, which is always the opposite sign to the number printed inside the bracket.

How do you find the turning point of a quadratic in vertex form?

Set the bracket equal to zero and solve for x. For y equals (x plus p) squared plus q, set x plus p equals zero to get x equals negative p. The y-coordinate is q directly, since the square term is zero at the turning point. The turning point is (negative p, q).

Why does the sign of the x-coordinate differ from the sign inside the bracket?

Because you are solving an equation, not reading a value. In (x minus 7), setting the bracket to zero gives x minus 7 equals zero, so x equals positive 7. In (x plus 5), setting x plus 5 equals zero gives x equals negative 5. The sign reverses because you are solving for x, not substituting it.

GCSE Maths Higher · AQA · Completing the square and turning points

Turning point sign from vertex form: reading x from the bracket

When students see $y = (x-7)^2+8$ , the instinct is to read the $-7$ from inside the bracket and write the turning point as $(-7,\,8)$ . That is wrong. The turning point is where the squared term equals zero. Setting $(x-7)=0$ gives $x=7$ , not $x=-7$ . At that point, $y=0+8=8$ , so the turning point is $(7,\,8)$ .

The rule is: the x-coordinate of the turning point is found by setting the bracket to zero and solving. It is always the opposite sign to the number printed inside the bracket. Reading the printed sign directly is one of the most consistent sign errors on AQA Higher completing-the-square questions.

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How to spot it in your own work

You wrote $(-7,\,8)$ for $y=(x-7)^2+8$ , copying the printed $-7$ from the bracket instead of solving $x-7=0$ to get $x=7$ .
You wrote $(5,\,-3)$ for $y=(x+5)^2-3$ , reading the +5 directly instead of solving $x+5=0$ to get $x=-5$ .
You wrote $(-9,\,-11)$ for $y=(x-9)^2-11$ , copying $-9$ from the bracket instead of solving to get $x=9$ .
You applied an inconsistent sign rule: you flipped the sign in one case but copied it in another, rather than always setting the bracket to zero.

An exam question that triggers it

Here is the structure of a typical AQA Higher turning-point question:

The curve C has equation $y = (x-9)^2-11$ .

Write down the coordinates of the turning point of C.

The misconception writes $(-9,\,-11)$ (copies the printed $-9$ ). The fix: set $x-9=0$ to get $x=9$ , read $q=-11$ directly. Turning point: $(9,\,-11)$ .

Why students fall for this

The vertex form $y=(x+p)^2+q$ makes the turning point visually accessible: the two numbers $p$ and $q$ appear explicitly in the expression. Students pick up $q$ correctly as the y-coordinate (it is unambiguous), but they read $p$ as the x-coordinate without solving the bracket equation. Because $p$ appears with its sign printed inside the bracket, copying it directly gives the wrong sign for x.

A second pattern appears when students know the sign should change but apply a memorised flip rule inconsistently: they flip when the bracket has a minus but copy when it has a plus (or vice versa), because they have not internalised the reason. The bracket-to-zero method handles both cases systematically: $x-7=0$ gives $x=+7$ , and $x+5=0$ gives $x=-5$ .

The fix: Set the bracket to zero and solve: x-coordinate of the turning point is the solution to (bracket) = 0

Step 1: identify the bracket. In $y=(x-9)^2-11$ , the bracket is $(x-9)$ .

Step 2: set the bracket to zero. $x-9=0$ .

Step 3: solve for x. $x=9$ .

Step 4: read the y-coordinate. At the turning point the bracket is zero, so $y=0+(-11)=-11$ . The constant $q$ is the y-coordinate directly.

Step 5: write the turning point. $(9,\,-11)$ .

Step 6: verify by substitution. At $x=9$ : $y=(9-9)^2-11=0-11=-11$ ✓. At the trap $x=-9$ : $y=(-18)^2-11=324-11=313$ , which is not $-11$ , confirming $(-9,\,-11)$ is wrong.

Worked example

Find the turning point of $y = (x-7)^2+8$ .

Set bracket to zero. $x-7=0 \Rightarrow x=7$ .
Read the y-coordinate. $y=0+8=8$ .
Turning point.
$(7,\,8)$
Trap: $(-7,\,8)$ gives $y=(-14)^2+8=204$ at $x=-7$ , confirming it is wrong.

Find the turning point of $y = (x+5)^2-3$ .

Set bracket to zero. $x+5=0 \Rightarrow x=-5$ .
Read the y-coordinate. $y=0+(-3)=-3$ .
Turning point.
$(-5,\,-3)$
Trap: $(5,\,-3)$ . At $x=5$ : $y=(5+5)^2-3=100-3=97$ , which is not $-3$ .

Find the turning point of $y = (x-9)^2-11$ .

Set bracket to zero. $x-9=0 \Rightarrow x=9$ .
Read the y-coordinate. $y=0+(-11)=-11$ .
Turning point.
$(9,\,-11)$
Verify: $(9-9)^2-11=0-11=-11$ ✓

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Common questions

Why is the turning point of $y=(x-7)^2+8$ at $x=7$ and not $x=-7$ ?: Because the turning point is where the squared bracket equals zero. Setting $x-7=0$ gives $x=7$ . Substituting $x=-7$ gives $y=(-7-7)^2+8=196+8=204$ , not $8$ , so $(-7,8)$ is not the turning point.
How does the sign rule differ between $(x-7)^2$ and $(x+5)^2$ ?: In both cases you set the bracket to zero and solve. For $(x-7)=0$ : $x=+7$ (bracket has minus, vertex x is positive). For $(x+5)=0$ : $x=-5$ (bracket has plus, vertex x is negative). The outcomes differ but the method is the same: solve the bracket equation.
Is there a quick formula for the turning point of $y=(x+p)^2+q$ ?: Yes: the turning point is $(-p,\,q)$ . But this formula comes from solving $x+p=0$ to get $x=-p$ . If you remember the formula as a fact rather than as a result of solving the bracket, you are likely to apply it inconsistently when the sign inside changes. The safe habit is always to set the bracket to zero and solve.

Related misconceptions

Completing the square: forgetting to subtract the adjustment constantKeeping the original constant unchanged when completing the square, writing (x+4)²−5 instead of (x+4)²−21, a companion misconception on the same Higher topic.
Negative indices mean reciprocals, not negative numbersReading the minus in 8^-5 as a sign on the result and writing −8^5, when the correct value is 1/8^5, a different Higher topic with the same trap structure of misreading a sign.

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