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Q6 · Explain the parameters and vertex of y = a(x − h)² + k

Learn to explain the parameters and vertex of y = a(x − h)² + k through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Quadratic Relations

Understanding the Parameters and Vertex

In Grade 9, you worked with the basic quadratic relation y = x². You saw that its graph is a U-shaped curve called a parabola, that it has a turning point at the origin, and that it is symmetric about the y-axis. In this lesson you will learn a powerful way of writing any quadratic relation — vertex form — that makes it easy to read off the turning point and to see exactly how the graph has been transformed from y = x². Every feature of vertex form connects to something visual on the graph, so understanding it gives you real control over how parabolas behave.

What you will learn

Identify the values of a, h, and k in the vertex form y = a(x − h)² + k and describe what each one does to the parabola.
State the coordinates of the vertex directly from the equation in vertex form.
Describe how changing a affects the direction of opening and the vertical stretch or compression of the parabola.
Describe how changing h and k shifts the parabola horizontally and vertically.
Write the equation of a parabola in vertex form when given its vertex and one other point.

Reviewing the Base Parabola y = x²

Before exploring vertex form, it helps to be clear about the parent parabola y = x². Every point on this curve is found by squaring an x-value. For example, when x = 2, y = 4, and when x = −2, y = 4 as well. This symmetry means the left and right sides of the parabola are mirror images of each other.

The turning point — where the parabola changes from falling to rising — is called the vertex. For y = x², the vertex sits at the origin, (0, 0). The vertical line that passes through the vertex is called the axis of symmetry. For y = x² that line is x = 0, the y-axis.

All of the transformations in this lesson will move or reshape y = x² in predictable ways. The vertex form equation is the tool that describes exactly how.

The parent parabola y = x² has its vertex at (0, 0).
The axis of symmetry is the vertical line through the vertex.
The parabola opens upward when y-values increase away from the vertex.

Introducing Vertex Form and Its Three Parameters

Vertex form is written as y = a(x − h)² + k. The letters a, h, and k are called parameters. A parameter is a number in an equation that you can change in order to produce different, but related, curves. Each parameter controls a specific feature of the parabola.

The parameter a is the coefficient in front of the squared bracket. It controls two things at once: the direction in which the parabola opens, and how wide or narrow it looks. When a is positive, the parabola opens upward. When a is negative, it opens downward — picture a hill instead of a valley. The size of a (ignoring its sign) tells you about stretch: if |a| is greater than 1, the parabola is narrower than y = x²; if |a| is between 0 and 1, it is wider.

The parameter h controls a horizontal shift. Notice the bracket is written as (x − h), not (x + h). This means if h = 3, the bracket reads (x − 3) and the parabola shifts 3 units to the right — not to the left. This trips many students up, so always read h directly from the equation with its sign: whatever number is being subtracted from x inside the bracket is h.

The parameter k controls a vertical shift. Adding k outside the squared bracket raises the entire parabola by k units when k is positive, and lowers it by |k| units when k is negative. Unlike h, k works exactly as you might expect: positive k moves the graph up, negative k moves it down.

Because h and k together give the location of the vertex, vertex form makes it very easy to read the vertex without doing any calculation. The vertex is always the point (h, k).

y = a(x - h)^2 + k

a controls direction of opening (positive → up, negative → down) and width.
|a| > 1 gives a narrower parabola; 0 < |a| < 1 gives a wider parabola.
h is the horizontal shift: (x − h) shifts the vertex h units to the right when h > 0.
k is the vertical shift: positive k moves the vertex up, negative k moves it down.
The vertex is at (h, k) and the axis of symmetry is x = h.

Seeing Each Parameter in Action

It helps to change one parameter at a time and observe what happens. Start with y = x² and change only a. Comparing y = 2x², y = x², and y = 0.5x² shows parabolas that all share the same vertex at (0, 0) but have different widths. The graph of y = 2x² is narrower because y-values grow more quickly as x moves away from zero. The graph of y = 0.5x² is wider because y-values grow more slowly.

Now change only k. The equation y = x² + 3 keeps the same shape as y = x² but the vertex climbs to (0, 3). The equation y = x² − 5 drops the vertex to (0, −5). The shape and width are unchanged.

Now change only h. The equation y = (x − 4)² has its vertex at (4, 0). It is the same shape as y = x², but the whole parabola has slid 4 units to the right. The equation y = (x + 2)² can be rewritten as y = (x − (−2))², so h = −2 and the vertex is at (−2, 0) — a shift of 2 units to the left.

Combining all three: y = −3(x − 1)² + 7 has a = −3, h = 1, and k = 7. Its vertex is at (1, 7), it opens downward, and it is narrower than y = x².

Changing only a reshapes the parabola without moving the vertex.
Changing only k slides the parabola straight up or down.
Changing only h slides the parabola left or right.
y = (x + 2)² is the same as y = (x − (−2))², so h = −2, not h = 2.

Reading the Vertex and Axis of Symmetry

The vertex is the most important single point of a parabola. In vertex form, you can read it immediately: the vertex is (h, k). You do not need to substitute any values or solve any equations.

The axis of symmetry is the vertical line that passes through the vertex. Because the vertex has x-coordinate h, the axis of symmetry is the line x = h. Every point on the parabola has a mirror-image point on the other side of this line at the same height.

The vertex is also the minimum point of the parabola when a > 0 (the lowest point of the valley), and the maximum point when a < 0 (the top of the hill). Knowing whether the vertex is a minimum or maximum depends entirely on the sign of a.

(h, k)

Vertex = (h, k), read directly from the equation.
Axis of symmetry: x = h.
When a > 0, the vertex is a minimum; when a < 0, the vertex is a maximum.

Writing an Equation in Vertex Form

Sometimes you are given information about a parabola and need to build the equation yourself. The most common situation is: you know the vertex and one other point on the curve. Here is the process.

Step 1: Substitute h and k from the vertex directly into y = a(x − h)² + k. At this point, a is the only unknown.

Step 2: Substitute the coordinates of the other known point for x and y in the equation. Now the equation has only one unknown, a.

Step 3: Solve for a using algebra you learned in Grade 9.

Step 4: Write the final equation with all three values filled in. This four-step process appears in both worked examples below, so you will see it applied to two different situations.

If you know the vertex (h, k), place those values into vertex form immediately.
A second known point lets you solve for a by substitution.
Always verify your answer by substituting the known point back into the final equation.

What Each Parameter Does to y = a(x − h)² + k

Parameter	What you change	Effect on the graph	Example
a (positive)	Sign of a	Parabola opens upward	a = 3 → opens up
a (negative)	Sign of a	Parabola opens downward	a = −3 → opens down
\|a\| > 1	Size of \|a\|	Parabola is narrower than y = x²	a = 4 → narrow
0 < \|a\| < 1	Size of \|a\|	Parabola is wider than y = x²	a = 0.25 → wide
h > 0	Value of h	Vertex shifts right by h units	h = 5 → vertex at x = 5
h < 0	Value of h	Vertex shifts left by \|h\| units	h = −5 → vertex at x = −5
k > 0	Value of k	Vertex shifts up by k units	k = 6 → vertex at y = 6
k < 0	Value of k	Vertex shifts down by \|k\| units	k = −6 → vertex at y = −6

Worked example

Reading Parameters and Sketching Key Features

A parabola has equation y = −2(x − 3)² + 8. State the values of a, h, and k. Then give the vertex, the axis of symmetry, and state whether the vertex is a minimum or a maximum. Finally, find the y-intercept.

Identify a, h, and k
Compare the equation to the template $y = a(x - h)^2 + k$ . The number multiplying the squared bracket is $a$ . The number subtracted inside the bracket is $h$ . The number added outside is $k$ . Reading directly from $y = -2(x - 3)^2 + 8$ : $a = -2$ , $h = 3$ , $k = 8$ .
$a = -2, h = 3, k = 8$
State the vertex and axis of symmetry
The vertex is always the point $(h, k)$ , so here the vertex is $(3, 8)$ . The axis of symmetry is the vertical line $x = h$ , so the axis of symmetry is $x = 3$ .
$\text{vertex} = (3, 8), x = 3$
Determine minimum or maximum
Because $a = -2$ , which is negative, the parabola opens downward. A downward-opening parabola has a highest point, so the vertex is a maximum.
$a < 0 \Rightarrow \text{maximum at } (3, 8)$
Find the y-intercept
The y-intercept is the point where the graph crosses the y-axis, which is where $x = 0$ . Substitute $x = 0$ into the equation: $y = -2(0 - 3)^2 + 8 = -2(9) + 8 = -18 + 8 = -10$ . So the y-intercept is $(0, -10)$ .
$y = -2(0-3)^2 + 8 = -2(9) + 8 = -10$

Answer: a = −2, h = 3, k = 8; vertex (3, 8); axis of symmetry x = 3; the vertex is a maximum; y-intercept (0, −10).

Check: Substitute the y-intercept back: y = −2(0 − 3)² + 8 = −2(9) + 8 = −18 + 8 = −10. ✓ The y-intercept checks out.

Worked example

Building a Vertex Form Equation from a Vertex and a Point

A parabola has its vertex at (−1, 5) and passes through the point (2, −13). Write the equation of the parabola in vertex form.

Set up vertex form with h and k
The vertex is $(-1, 5)$ , so $h = -1$ and $k = 5$ . Substitute these into $y = a(x - h)^2 + k$ . Remember that subtracting $h = -1$ gives a plus sign inside the bracket.
$y = a(x - (-1))^2 + 5 = a(x + 1)^2 + 5$
Substitute the known point to create an equation in a
The parabola passes through $(2, -13)$ , meaning $x = 2$ gives $y = -13$ . Substitute both values into the equation from Step 1 so that $a$ is the only unknown.
$-13 = a(2 + 1)^2 + 5$
Simplify the bracket and isolate a
Calculate $(2 + 1)^2 = 3^2 = 9$ . The equation becomes $-13 = 9a + 5$ . Subtract 5 from both sides to get $-18 = 9a$ , then divide both sides by 9 to find $a = -2$ .
$-18 = 9a \Rightarrow a = -2$
Write the final equation
Now that all three parameters are known — $a = -2$ , $h = -1$ , $k = 5$ — substitute them into vertex form to get the complete equation.
$y = -2(x + 1)^2 + 5$

Answer: y = −2(x + 1)² + 5

Check: Substitute (2, −13): y = −2(2 + 1)² + 5 = −2(9) + 5 = −18 + 5 = −13. ✓ The point lies on the parabola.

Common mistakes and how to avoid them

Reading h as the number written in the bracket, ignoring the minus sign. For example, seeing y = (x − 4)² and saying h = −4.

Correction: The template is y = a(x − h)², so h is the number subtracted from x. In y = (x − 4)², h = 4 (positive). In y = (x + 3)², rewrite as y = (x − (−3))², so h = −3.

Thinking a wider parabola has a larger value of |a|.

Correction: A larger |a| makes the parabola narrower, because y-values grow faster. A smaller |a| (between 0 and 1) makes it wider.

Placing the vertex at (k, h) instead of (h, k).

Correction: The vertex is always (h, k): h is the x-coordinate and k is the y-coordinate, matching the order in y = a(x − h)² + k.

Forgetting to square the bracket when finding the y-intercept. For example, computing y = −2(0 − 3) + 8 = −6 + 8 = 2 instead of y = −2(0 − 3)² + 8 = −10.

Correction: Always apply the exponent first. (0 − 3)² = (−3)² = 9, not −3. Follow the order of operations: brackets first, then exponents.

When solving for a, substituting the vertex coordinates instead of the second known point.

Correction: The vertex coordinates were already used to set h and k. The second point is the one you substitute for x and y to solve for a.

Lesson summary

Vertex form y = a(x − h)² + k has three parameters: a, h, and k, each controlling a different feature of the parabola.
The vertex of the parabola is at (h, k) and can be read directly from the equation without any calculation.
The axis of symmetry is the vertical line x = h.
The sign of a tells you the direction of opening: positive a means the parabola opens upward (vertex is a minimum); negative a means it opens downward (vertex is a maximum).
The size of |a| controls the width: |a| > 1 gives a narrower parabola, and 0 < |a| < 1 gives a wider one.
To write a vertex form equation, substitute h and k from the vertex, then substitute a second known point to solve for a.

Check your understanding

Question 1

What is the vertex of the parabola y = 4(x − 7)² − 3?

(−7, −3)
(7, 3)
(7, −3)
(−7, 3)

Show answer and explanation

(7, −3)

The vertex is (h, k). Comparing to y = a(x − h)² + k, h = 7 (the number subtracted from x inside the bracket) and k = −3 (the number added outside). So the vertex is (7, −3).

Question 2

A parabola has the equation y = −0.5(x + 2)² + 6. Which statement is correct?

The parabola opens upward and the vertex is a minimum.
The parabola opens downward and is wider than y = x².
The parabola opens upward and is narrower than y = x².
The parabola opens downward and is narrower than y = x².

Show answer and explanation

The parabola opens downward and is wider than y = x².

Here a = −0.5. Because a is negative, the parabola opens downward. Because |a| = 0.5, which is between 0 and 1, the parabola is wider than y = x². So the correct statement is: opens downward and wider than y = x².

Question 3

A parabola in vertex form has vertex (−3, 4) and passes through (0, −14). What is the value of a?

a = −2
a = 2
a = −18
a = 6

Show answer and explanation

a = −2

Set up y = a(x − (−3))² + 4 = a(x + 3)² + 4. Substitute (0, −14): −14 = a(0 + 3)² + 4 = 9a + 4. Then 9a = −18, so a = −2.

Question 4

Which equation represents a parabola with vertex (2, −1) that opens upward and is narrower than y = x²?

y = 0.3(x − 2)² − 1
y = −3(x − 2)² − 1
y = 3(x − 2)² − 1
y = 3(x + 2)² − 1

Show answer and explanation

y = 3(x − 2)² − 1

The vertex (2, −1) means h = 2 and k = −1, giving y = a(x − 2)² − 1. Opening upward requires a > 0. Narrower than y = x² requires |a| > 1. Only y = 3(x − 2)² − 1 satisfies all three conditions.

Key terms

Parabola: The U-shaped (or hill-shaped) curve that is the graph of any quadratic relation.
Vertex: The turning point of a parabola — the lowest point when it opens upward or the highest point when it opens downward.
Vertex form: The way of writing a quadratic relation as y = a(x − h)² + k, where (h, k) is the vertex.
Parameter: A number in an equation that can be changed to produce different but related curves. In vertex form the parameters are a, h, and k.
Axis of symmetry: The vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is x = h.
Vertical stretch: A transformation that makes the parabola narrower by multiplying all y-values by a factor greater than 1, controlled by |a| > 1.
Vertical compression: A transformation that makes the parabola wider by multiplying all y-values by a factor between 0 and 1, controlled by 0 < |a| < 1.
y-intercept: The point where the graph crosses the y-axis, found by substituting x = 0 into the equation.

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