Identify the vertex, axis, intercepts, and extrema of a parabola

Ontario Grade 10 Mathematics

Quadratic Relations

Finding the vertex, axis, intercepts, and extrema from a graph

In Grade 9, you worked with graph coordinates and learned to read points from a grid. A point is written as an ordered pair: the first number gives its horizontal position, and the second gives its vertical position. For example,

(3,-2)

is three units right and two units below the origin. In this lesson, you will use those skills to describe important features of a parabola. You will interpret a supplied graph or a graph made with technology. No equation is needed: the graph provides the information.

What you will learn

Read a parabola’s vertex and axis of symmetry from a graph.
Identify the x- and y-intercepts as points.
Decide whether the vertex is a maximum or minimum.
Describe the extrema using the graph’s direction and coordinates.

Start with the shape and the vertex

A parabola is a curved graph with two sides that mirror each other. It may open upward, like a bowl, or downward, like an upside-down bowl. Look at the graph before reading any coordinates. The direction it opens helps you decide whether its turning point is a lowest or highest point.

The vertex is the turning point of the parabola. It is where the curve changes direction. Read its position from the grid as an ordered pair, with the horizontal coordinate first and the vertical coordinate second. If the vertex is at

(2,-3)

, for instance, the graph turns at horizontal position

2

and vertical position

-3

.

A vertical line through the vertex divides the two sides of the parabola into matching halves. This is the axis of symmetry. Its equation uses the vertex’s horizontal coordinate. If the vertex has horizontal coordinate

2

, the axis is

x=2

. The line is vertical because every point on it has the same

x

-coordinate.

x=h

The vertex is the turning point.
The axis of symmetry is a vertical line through the vertex.
Use the vertex’s first coordinate to write the axis.

Read the intercepts from the axes

An intercept is a point where the graph meets one of the coordinate axes. The horizontal axis is the

x

-axis; the vertical axis is the

y

-axis. Read the point where the curve meets an axis, then write both coordinates in order.

An

x

-intercept is where the curve crosses or touches the horizontal axis. Every point on that axis has a vertical coordinate of

0

, so an

x

-intercept has the form

(x,0)

. A parabola can meet the

x

-axis twice, once, or not at all. Count only the points shown by the graph.

The

y

-intercept is where the curve meets the vertical axis. Every point on that axis has a horizontal coordinate of

0

, so a

y

-intercept has the form

(0,y)

. A parabola has one

y

-intercept. If the graph meets the vertical axis at height

4

, write the point as

(0,4)

.

Sometimes the graph’s symmetry helps check what you read. When there are two

x

-intercepts, the axis of symmetry lies halfway between them. Use this as a visual check, not as a replacement for reading the vertex from the graph.

At an $x$ -intercept, $y=0$ .
At a $y$ -intercept, $x=0$ .
Write intercepts as points, not just as single numbers.

Find the extremum

An extremum is the greatest or least vertical value reached by a graph. The word extrema means more than one extremum. For a parabola, the extremum occurs at the vertex.

If the parabola opens upward, its vertex is the lowest point. The vertical coordinate of the vertex is the minimum value. For example, a vertex at

(2,-3)

on an upward-opening graph gives a minimum value of

-3

, occurring at

x=2

.

If the parabola opens downward, its vertex is the highest point. Its vertical coordinate is the maximum value. For a vertex at

(-1,5)

on a downward-opening graph, the maximum value is

5

, occurring at

x=-1

.

Be precise about the wording. The extremum is a vertical value, not a coordinate pair. You can also state where it occurs by giving the vertex’s horizontal coordinate.

An upward-opening parabola has a minimum at its vertex.
A downward-opening parabola has a maximum at its vertex.
The extremum is the vertex’s vertical coordinate.

A reliable graph-reading routine

First, check whether the graph opens upward or downward. This tells you whether the vertex is a minimum or maximum. Next, locate the turning point and read its coordinates. Use the first coordinate to write the axis of symmetry, and use the second coordinate to name the extremum.

Then inspect where the curve meets each axis. Read every meeting point on the

x

-axis and write it as an ordered pair with vertical coordinate

0

. Find where it meets the

y

-axis and write that point with horizontal coordinate

0

. Finally, check that the vertex, axis, and intercepts agree with the visible shape of the graph.

When a graph is drawn on a grid, check the scale on both axes before reading. Each square may represent one unit, or it may represent a different amount. If a point lies between grid marks, report only the precision the graph supports.

Observe the opening direction before naming the extremum.
Read coordinates in horizontal-then-vertical order.
Check the axis and intercepts against the shape and scale of the graph.

What to look for on the graph

Feature	What to identify	How to write it
Vertex	Turning point	$(h,k)$
Axis of symmetry	Vertical line through the vertex	$x=h$
$x$ -intercept	Point where the graph meets the horizontal axis	$(x,0)$
$y$ -intercept	Point where the graph meets the vertical axis	$(0,y)$
Extremum	Highest or lowest vertical value	Maximum or minimum value $k$

Worked example

Example 1: An upward-opening graph

A supplied graph opens upward. Its turning point is at

(-2,-9)

. It meets the

x

-axis at

(-5,0)

and

(1,0)

, and it meets the

y

-axis at

(0,-5)

. Identify the vertex, axis of symmetry, intercepts, and extremum.

Read the vertex
The turning point is the vertex. The graph shows that point at horizontal coordinate $-2$ and vertical coordinate $-9$ .
$(-2,-9)$
Write the axis
The axis is the vertical line through the vertex. Use the vertex’s horizontal coordinate, so the axis is $x=-2$ .
$x=-2$
Record the intercepts
The graph meets the horizontal axis at the two listed points, so these are its $x$ -intercepts. It meets the vertical axis at the listed $y$ -intercept.
$(-5,0),\ (1,0),\ (0,-5)$
Name the extremum
The graph opens upward, so the vertex is its lowest point. Its vertical coordinate is $-9$ , giving a minimum value of $-9$ at $x=-2$ .
Check the symmetry
The two $x$ -intercepts are equally far from the axis: each is $3$ units away horizontally. This supports the axis reading.

Answer: Vertex:

(-2,-9)

. Axis of symmetry:

x=-2

. The

x

-intercepts are

(-5,0)

and

(1,0)

, and the

y

-intercept is

(0,-5)

. The minimum value is

-9

, occurring at

x=-2

.

Check: The curve opens upward, so its vertex must be a minimum. The two

x

-intercepts lie equally far from

x=-2

, which agrees with the graph’s symmetry.

Worked example

Example 2: A downward-opening graph touching the axis

A technology-generated graph opens downward. Its turning point is

(4,0)

, and it meets the

y

-axis at

(0,-16)

. The graph shows no other meeting point with the

x

-axis. Identify the vertex, axis of symmetry, intercepts, and extremum.

Identify the vertex
The turning point shown on the graph is the vertex. It lies on the horizontal axis, so its vertical coordinate is $0$ .
$(4,0)$
State the axis
The axis is vertical and passes through the vertex. Its horizontal coordinate is $4$ .
$x=4$
Read the intercepts
The graph touches the $x$ -axis at its vertex and shows no other point on that axis. Therefore there is one $x$ -intercept. The supplied point on the $y$ -axis is the $y$ -intercept.
$(4,0),\ (0,-16)$
Name the extremum
A downward-opening graph has its highest point at the vertex. The vertical coordinate there is $0$ , so the maximum value is $0$ , occurring at $x=4$ .

Answer: Vertex:

(4,0)

. Axis of symmetry:

x=4

. The only

x

-intercept is

(4,0)

, and the

y

-intercept is

(0,-16)

. The maximum value is

0

, occurring at

x=4

.

Check: The vertex lies on the

x

-axis, so it is an

x

-intercept. Since the graph opens downward from that highest point, the graph is below the axis on either side, consistent with one point of contact.

Common mistakes and how to avoid them

Writing the axis of symmetry as a number, such as

-2

.

Correction: The axis is a vertical line, so write an equation such as

x=-2

.

Reversing the coordinates of a point.

Correction: Write the horizontal coordinate first and the vertical coordinate second. An

x

-intercept has the form

(x,0)

; a

y

-intercept has the form

(0,y)

.

Calling the vertical coordinate of the vertex the location of the extremum.

Correction: The extremum is the maximum or minimum value. Also state the horizontal coordinate where it occurs.

Assuming every graph crosses the

x

-axis twice.

Correction: Count the actual points where the graph meets the axis. It may meet it twice, once, or not at all.

Calling the vertex a maximum on an upward-opening graph.

Correction: An upward-opening graph has a lowest point at its vertex, so its extremum is a minimum.

Lesson summary

The vertex is the turning point of the parabola.
The axis of symmetry is the vertical line through the vertex and uses its horizontal coordinate.
An $x$ -intercept lies on the horizontal axis; a $y$ -intercept lies on the vertical axis.
The extremum is the vertex’s vertical coordinate: a minimum when the graph opens upward and a maximum when it opens downward.
Read and report features from the graph, paying attention to the grid scale.

Check your understanding

Question 1

A graph has vertex

(-3,5)

and opens downward. What is its axis of symmetry?

$y=5$
$x=-3$
$x=5$
$y=-3$

Show answer and explanation

x=-3

The axis is vertical and passes through the vertex, so it uses the vertex’s horizontal coordinate:

x=-3

.

Question 2

A graph opens upward and has vertex

(1,-4)

. What is its extremum?

Maximum value of $1$
Maximum value of $-4$
Minimum value of $-4$
Minimum value of $1$

Show answer and explanation

Minimum value of

-4

An upward-opening graph has a minimum at its vertex. The vertical coordinate is

-4

, so the minimum value is

-4

.

Question 3

A graph touches the

x

-axis at its vertex

(2,0)

and has no other point on that axis. How many

x

-intercepts does it have?

None
Exactly one
Exactly two
It cannot be read from the graph

Show answer and explanation

Exactly one

The single point where the graph touches the axis is an

x

-intercept. The graph shows no second point on the axis.

Key terms

Parabola: A curved graph with two sides that mirror each other and a turning point.
Vertex: The turning point of a parabola, written as an ordered pair.
Axis of symmetry: The vertical line through the vertex that divides the parabola into matching halves.
Intercept: A point where a graph meets one of the coordinate axes.
$x$ -intercept: A point where the graph meets the horizontal axis; its vertical coordinate is $0$ .
$y$ -intercept: A point where the graph meets the vertical axis; its horizontal coordinate is $0$ .
Extremum: The greatest or least vertical value reached by a graph.

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