Interpret information from standard, vertex, and factored forms

Ontario Grade 11 Mathematics

Quadratic Functions

Interpret standard, vertex, and factored forms

A quadratic function has a graph shaped like a U or an upside-down U. The same quadratic can be written in different ways. Each way makes some information easier to see. For example, a factored equation can show where the graph crosses the horizontal axis, while a vertex equation shows its turning point. In this lesson, you will learn to read those clues and connect them to the graph.

What you will learn

Recognize standard, vertex, and factored forms of a quadratic function.
Identify what each form directly tells you about a graph.
Use features from one form to interpret the same quadratic function in another form.

1. Prerequisite bridge: coordinates and intercepts

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

(2, 5)

. The first number is the

x

-coordinate, and the second is the

y

-coordinate. A function gives a

y

-value for each allowed

x

-value.

The

y

-intercept is where a graph crosses the vertical axis. At that point,

x=0

. An

x

-intercept is where the graph crosses the horizontal axis. At an

x

-intercept,

y=0

. These ideas help us interpret quadratic equations.

A quadratic function has an

x^2

term and can be written in several forms. The coefficient

a

is nonzero. Its graph is called a parabola. If

a

is positive, the parabola opens upward. If

a

is negative, it opens downward.

a \ne 0

An intercept is a point where a graph meets an axis.
The sign of $a$ tells whether the parabola opens upward or downward.

2. What each form makes easy to read

Standard form is

y=ax^2+bx+c

. The coefficients are numbers that multiply the powers of

x

. The constant term,

c

, gives the

y

-intercept because setting

x=0

leaves

y=c

. The coefficient

a

tells the opening direction. Its size also affects how narrow or wide the parabola looks compared with

y=x^2

.

Vertex form is

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

. The vertex is the turning point of the parabola, written as

(h,k)

. It is the lowest point when the graph opens upward and the highest point when it opens downward. The line

x=h

is the axis of symmetry: it divides the parabola into matching left and right sides. The value of

a

again tells the opening direction and affects the graph’s width.

Factored form is

y=a(x-r)(x-s)

. The factors show the zeros, also called the roots or

x

-intercepts. The graph crosses or touches the horizontal axis at

(r,0)

and

(s,0)

. If the two values are equal, there is one repeated zero. The axis of symmetry lies halfway between the two zeros.

All three forms describe the same kind of function. No form shows every feature equally directly. Read the form first, then decide which information is visible and which information must be worked out.

y=ax^2+bx+c,\quad y=a(x-h)^2+k,\quad y=a(x-r)(x-s)

Standard form directly shows the $y$ -intercept.
Vertex form directly shows the vertex and axis of symmetry.
Factored form directly shows the zeros or $x$ -intercepts.
The coefficient $a$ appears in all three forms and controls opening direction.

3. Connect the forms to the graph

Use a sketch or graphing tool to check your interpretation. In vertex form, plot the vertex first. Then use the opening direction to sketch the curve on both sides of the axis of symmetry. In factored form, plot the zeros first. The midpoint between them gives the axis of symmetry. In standard form, plot the

y

-intercept and use the opening direction as a starting point.

For a quadratic with two zeros, the vertex lies halfway between their

x

-coordinates. For example, if the zeros are

-3

and

1

, their midpoint is

-1

. This tells you the axis of symmetry is

x=-1

. You still need the function’s value at that

x

-coordinate to find the vertex’s height.

The following table summarizes what each form reveals most quickly. The forms do not change the function; they change which features are easiest to see.

x_{\text{axis}}=\frac{r+s}{2}

Use the form as a source of clues, not as a different function.
A graph or table can help confirm that the clues fit together.

4. Guided example and application

Consider a quadratic written in factored form. We can identify its zeros and axis of symmetry directly. Then we can find its vertex and write equivalent vertex and standard forms. This shows how information from one form can help interpret the others.

In an application, the variables might represent time and height, or distance and cost. The equation’s context tells you what the axes and intercepts mean. The algebraic features still have the same roles: zeros occur where the output is zero, and the vertex is a maximum or minimum point. Interpret these values in context only when the question supplies that context.

The zeros give the horizontal-axis crossings.
The vertex gives the maximum or minimum value and its input.

What each form reveals

Form	Equation pattern	Information shown directly
Standard	$y=ax^2+bx+c$	$y$ -intercept $(0,c)$ ; opening direction from $a$
Vertex	$y=a(x-h)^2+k$	Vertex $(h,k)$ ; axis of symmetry $x=h$
Factored	$y=a(x-r)(x-s)$	Zeros $r$ and $s$ ; corresponding $x$ -intercepts

Worked example

Interpret and rewrite one quadratic

For

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

, identify the zeros, axis of symmetry, vertex, opening direction, and

y

-intercept. Then write the function in vertex and standard forms.

Read the zeros
A zero makes the output equal to zero. Each factor is zero at one of the horizontal-axis crossings. The factor $(x-1)$ is zero when $x=1$ , and $(x+3)$ is zero when $x=-3$ .
$x=1,\quad x=-3$
Find the axis
The axis of symmetry lies halfway between the zeros. Their midpoint is $-1$ , so the vertical line through the vertex is $x=-1$ .
$x=\frac{1+(-3)}{2}=-1$
Find the vertex height
Substitute the axis value into the function. This gives the output at the turning point. The vertex has coordinates $(-1,8)$ .
$y=-2(-1-1)(-1+3)=8$
Read the opening and intercept
The coefficient $a$ is $-2$ , so the parabola opens downward. To find the $y$ -intercept, set $x=0$ . The resulting point is $(0,6)$ .
$y=-2(0-1)(0+3)=6$
Write vertex form
Vertex form uses the vertex coordinates as $h$ and $k$ . With vertex $(-1,8)$ and the same coefficient $a=-2$ , the equation is shown below.
$y=-2(x+1)^2+8$
Expand to standard form
Expand the two factors, then multiply by $-2$ . The result is standard form, where the constant term also confirms the $y$ -intercept is $6$ .
$y=-2(x^2+2x-3)=-2x^2-4x+6$

Answer: The zeros are

x=-3

and

x=1

. The axis of symmetry is

x=-1

, and the vertex is

(-1,8)

. The parabola opens downward, and its

y

-intercept is

(0,6)

. Vertex form is

y=-2(x+1)^2+8

, and standard form is

y=-2x^2-4x+6

.

Check: The vertex form gives

y=8

at

x=-1

. The standard form gives

y=6

at

x=0

. Both agree with the values found from the factored form.

Common mistakes and how to avoid them

Reading the vertex in

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

as

(-h,k)

.

Correction: The equation uses

(x-h)

, so the vertex’s

x

-coordinate is

h

. If the bracket is

(x+4)

, then

h=-4

.

Treating the numbers in the factors as the zeros without checking their signs.

Correction: Set each factor equal to zero. For

(x-5)

, the zero is

x=5

; for

(x+5)

, it is

x=-5

.

Calling

c

the

y

-intercept value and forgetting the point.

Correction: In standard form, the intercept value is

c

, and the point is

(0,c)

.

Assuming a negative

a

changes the zeros in factored form.

Correction: The zeros come from the factors. The sign of

a

controls whether the parabola opens up or down.

Lesson summary

Standard form makes the $y$ -intercept easy to read.
Vertex form makes the vertex and axis of symmetry easy to read.
Factored form makes the zeros and $x$ -intercepts easy to read.
The sign of $a$ tells the opening direction in every form.
Use substitution, a midpoint, a table, or a graph to connect features across forms.

Check your understanding

Question 1

For

y=3(x-2)^2-5

, what is the vertex?

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

Show answer and explanation

(2,-5)

Compare with vertex form. The bracket is

(x-2)

, so

h=2

, and

k=-5

.

Question 2

For

y=-4(x+1)(x-6)

, what are the zeros?

$x=-1$ and $x=6$
$x=1$ and $x=-6$
$x=-4$ and $x=1$
$x=-1$ and $x=-6$

Show answer and explanation

x=-1

and

x=6

Set each factor equal to zero. The first gives

x=-1

, and the second gives

x=6

.

Question 3

What is the

y

-intercept of

y=2x^2-7x+4

?

$(0,4)$
$(4,0)$
$(0,-7)$
$(2,4)$

Show answer and explanation

(0,4)

In standard form,

c=4

. The

y

-intercept occurs at

x=0

, so its point is

(0,4)

.

Question 4

A quadratic has zeros

x=-2

and

x=8

. What is its axis of symmetry?

$x=3$
$x=6$
$x=-5$
$x=10$

Show answer and explanation

x=3

The axis is halfway between the zeros. Their midpoint is

3

, so the axis is

x=3

.

Key terms

Quadratic function: A function whose equation includes an $x^2$ term and whose graph is a parabola.
Vertex: The turning point of a parabola; it is a minimum or maximum point.
Axis of symmetry: The vertical line that divides a parabola into matching left and right sides.
Zero: An input value that makes the function’s output equal to zero.
Intercept: A point where a graph meets one of the coordinate axes.

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