A2.6 Sketch quadratic graphs from vertex form | Ontario Grade 11 Mathematics

Ontario Grade 11 Mathematics

Quadratic Functions

Read the vertex, use symmetry, and plot a few accurate points

A quadratic graph is a U-shaped curve or an upside-down U-shaped curve. Its shape is called a parabola. Vertex form gives a useful starting point for sketching because it shows the turning point directly. In this lesson, you will locate that point, decide which way the parabola opens, and plot matching points on both sides. A sketch should show the important features clearly; it does not need to include every possible point.

What you will learn

Recognize a quadratic equation written in vertex form.
Identify the vertex and axis of symmetry from the equation.
Use the coefficient and nearby points to decide the graph’s shape and sketch it.
Check that a sketch matches the equation.

1. Prerequisite bridge: coordinates and squared numbers

A point on a graph is written as an ordered pair, such as

(2, 5)

. The first number gives the horizontal position, called the

x

-coordinate. The second gives the vertical position, called the

y

-coordinate. To plot a point, move across to its

x

-coordinate and then up or down to its

y

-coordinate.

The square of a number is the number multiplied by itself. For instance,

3^2=9

and

(-3)^2=9

. Opposite numbers have the same square. This fact helps explain why quadratic graphs have matching points on either side of a central vertical line.

In vertex form, the expression inside the brackets tells how far an

x

-value is from a central

x

-coordinate. Squaring that distance gives the same result for equal distances to the left and right.

The first coordinate is horizontal; the second is vertical.
A positive or negative number has a positive square.
Equal horizontal distances from the centre give equal squared values.

2. Read the vertex form

The vertex form of a quadratic equation is

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

. The vertex is the turning point of the parabola. Its coordinates are

(h,k)

. The values

h

and

k

tell where the vertex is, while

a

tells the direction and steepness of the curve.

The axis of symmetry is a vertical line through the vertex. It divides the parabola into matching left and right sides. Its equation is

x=h

. The graph is symmetric because inputs the same distance from

h

make the squared part equal.

If

a

is positive, the parabola opens upward and the vertex is its lowest point. If

a

is negative, it opens downward and the vertex is its highest point. A larger value of |a| makes the graph narrower than the graph with

a=1

; a value of |a| between zero and one makes it wider. Here, |a| means the non-negative size of

a

without its sign.

Be careful with the sign inside the brackets. In

(x-4)^2

, the vertex has

x

-coordinate

4

. In

(x+4)^2

, the vertex has

x

-coordinate

-4

, because

x+4=x-(-4)

. The sign of

k

is read directly as the vertical coordinate.

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

Vertex: $(h,k)$ .
Axis of symmetry: $x=h$ .
The sign of $a$ gives the opening direction.
The size of |a| affects the width.

3. Build a sketch from points

Start by plotting the vertex. Draw or imagine the axis of symmetry through it. Then choose horizontal distances from the vertex and calculate the corresponding

y

-values. A small table helps keep the coordinates organized.

For a horizontal distance of one unit, the squared part is

1

. For a distance of two units, it is

4

. Multiply each result by

a

, then add

k

. This gives points on the right side and matching points on the left side. The graph should curve smoothly through them.

For example, if the vertex is at

(2,1)

and

a=1

, moving one unit from the vertex gives a vertical change of one unit. Moving two units gives a vertical change of four units. If

a

were negative, those changes would go downward instead of upward. This distance-based view connects the equation to the shape without needing to guess the curve.

A useful sketch includes the vertex, the axis of symmetry, enough plotted points to show the shape, and a smooth curve. Label the vertex when the sketch could otherwise be unclear. Do not connect the plotted points with straight line segments; the graph is curved.

Plot the vertex before choosing other points.
Use equal distances on both sides of the axis of symmetry.
Calculate points from the equation and draw a smooth curve through them.

4. Guided example and application

The worked example shows how to organize the information before drawing. Notice that the table includes points on both sides of the axis. These points are not separate guesses: symmetry means each pair has the same height.

Use vertex form to find the main features before plotting.
A table makes the symmetry visible.

Distance from the vertex and vertical change

Horizontal distance from vertex	Squared distance	Vertical change from vertex
0	0	0
1	1	a
2	4	4a

Worked example

Sketch a quadratic in vertex form

Sketch

y=-2(x-1)^2+3

. State the vertex, axis of symmetry, and opening direction, then find points one and two units from the vertex.

Read the vertex
Compare the equation with vertex form. The bracket is $(x-1)$ , so $h=1$ , and the final number gives $k=3$ . Therefore, the vertex is $(1,3)$ .
$(h,k)=(1,3)$
Find the axis and direction
The axis is the vertical line through the vertex. Since $a=-2$ is negative, the parabola opens downward, and the vertex is its highest point.
$x=1$
Calculate nearby points
Choose $x$ -values one and two units from $1$ . Substituting each value gives the matching $y$ -coordinate. Equal distances on opposite sides produce equal heights.
$\begin{array}{c|ccccc}x&-1&0&1&2&3\\\hline y&-5&1&3&1&-5\end{array}$
Draw the curve
Plot the vertex and the four calculated points. Draw a smooth, downward-opening curve through them. Check that the points at equal distances from $x=1$ have the same height.
$(-1,-5),\ (0,1),\ (1,3),\ (2,1),\ (3,-5)$

Answer: The sketch has vertex

(1,3)

, axis of symmetry

x=1

, and opens downward. It passes through

(0,1)

,

(2,1)

,

(-1,-5)

, and

(3,-5)

.

Check: The points at

x=0

and

x=2

are both one unit from the axis and have

y=1

. The points at

x=-1

and

x=3

are both two units from it and have

y=-5

. This confirms the symmetry.

Common mistakes and how to avoid them

Reading the vertex’s

x

-coordinate with the same sign as the number inside the brackets.

Correction: The bracket is

x-h

. For example,

(x+3)^2

can be read as

(x-(-3))^2

, so its vertex has

x

-coordinate

-3

.

Using the sign of

k

to decide which way the parabola opens.

Correction: The sign of

a

decides the opening direction. The value

k

moves the vertex up or down.

Plotting only points to the right of the vertex.

Correction: Plot matching points to the left as well. The two sides must have equal heights at equal distances from the axis.

Drawing straight lines between the plotted points.

Correction: Join the points with a smooth curve that opens in the direction indicated by

a

.

Lesson summary

Vertex form is $y=a(x-h)^2+k$ , with vertex $(h,k)$ and axis of symmetry $x=h$ .
A positive $a$ opens the parabola upward; a negative $a$ opens it downward.
Use distances from the vertex to calculate points, then reflect those points across the axis.
Plot the important points and draw a smooth curve through them.

Check your understanding

Question 1

For

y=3(x+2)^2-1

, what is the vertex?

$(2,-1)$
$(-2,-1)$
$(-2,1)$
$(3,-2)$

Show answer and explanation

(-2,-1)

The bracket

x+2

is

x-(-2)

, so

h=-2

. The final number is

k=-1

. The vertex is

(-2,-1)

.

Question 2

Which way does

y=-\frac{1}{2}(x-4)^2+2

open?

Upward, because the vertex is above the $x$ -axis
Downward, because the coefficient is negative
Upward, because the coefficient is between zero and one
Downward, because the vertex has positive coordinates

Show answer and explanation

Downward, because the coefficient is negative

The coefficient

a=-\frac{1}{2}

is negative, so the parabola opens downward. The vertex’s location does not determine the direction.

Question 3

A parabola has vertex

(1,2)

and axis

x=1

. If

(3,6)

lies on it, which other point must lie on it?

$(-1,6)$
$( -1,2)$
$(5,6)$
$(3,-6)$

Show answer and explanation

(-1,6)

The point

(3,6)

is two units to the right of the axis. Its matching point is two units to the left, at

x=-1

, with the same

y

-coordinate. So the point is

(-1,6)

.

Key terms

Quadratic: An equation whose graph is a parabola and whose variable includes a squared term.
Vertex: The turning point of a parabola.
Axis of symmetry: The vertical line that divides a parabola into matching left and right sides.
Parabola: The curved graph of a quadratic equation.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A2.6. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

A2.6 · Sketch quadratic graphs from vertex form

Ontario Grade 11 Mathematics

Read the vertex, use symmetry, and plot a few accurate points

What you will learn

1. Prerequisite bridge: coordinates and squared numbers

2. Read the vertex form

3. Build a sketch from points

4. Guided example and application

Distance from the vertex and vertical change

Sketch a quadratic in vertex form

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson