Sketch transformed linear, quadratic, square-root, and reciprocal functions

Ontario Grade 11 Mathematics

Characteristics of Functions

Using key points to sketch linear, quadratic, square-root, and reciprocal graphs

A graph can be shifted, stretched, compressed, or reflected without losing its basic shape. These changes are called transformations. You already know how to plot points and recognize a straight line, a parabola, and other familiar curves. In this lesson, you will use a few carefully chosen points from a parent function to sketch a transformed version. The four families are linear, quadratic, square-root, and reciprocal. A useful sketch shows the shape and the important features clearly; it does not need dozens of plotted points.

What you will learn

Recognize the parent functions for four familiar function families.
Describe how the parameters in a transformation change a graph.
Map useful points from a parent graph to sketch a transformed graph.
Show important features such as a vertex, endpoint, or reciprocal asymptotes.

Start with the four parent graphs

A parent function is the simplest version of a function family. It gives you a starting graph before any changes are made. The linear parent is

f(x)=x

: a straight line through the origin. The quadratic parent is

f(x)=x^2

: a U-shaped curve with its lowest point at the origin.

The square-root parent is

f(x)=\sqrt{x}

. It begins at the origin and extends to the right. Its input cannot be negative, so its graph has no points to the left of the vertical axis. The reciprocal parent is

f(x)=\frac{1}{x}

. It has one branch in the first quadrant and another in the third. It is not defined at

x=0

, and neither branch reaches either axis.

A key point is a point chosen because it helps you locate or shape a graph. The table gives useful points for each parent. For the reciprocal graph, the axes are asymptotes: lines the branches approach but do not reach. These parent points and features are the bridge from familiar graphs to transformed ones.

The four parent rules are $f(x)=x$ , $f(x)=x^2$ , $f(x)=\sqrt{x}$ , and $f(x)=\frac{1}{x}$ .
Use several key points to show a curve’s position and shape.
The reciprocal parent has asymptotes $x=0$ and $y=0$ .

Read the transformation and map points

A common transformation form is

y=a\,f(x-h)+k

. The function

f

tells you which parent shape to use. The value

h

shifts the graph horizontally: positive

h

moves it right, and negative

h

moves it left. The value

k

shifts the graph vertically: positive

k

moves it up, and negative

k

moves it down.

The value

a

changes the vertical position of points relative to the shifted graph. If

|a|>1

, the graph is vertically stretched; if

0<|a|<1

, it is vertically compressed. If

a<0

, it is also reflected across the

x

-axis. A reflection flips each point’s vertical displacement to the opposite side.

To map a parent point, add

h

to its

x

-coordinate and multiply its

y

-coordinate by

a

before adding

k

. This works because the input shift changes where the point sits horizontally, while the outside operations change its height. For example, the parent point

(1,2)

maps to

(1+h,2a+k)

.

(x,y)\mapsto(x+h,ay+k)

In $x-h$ , the horizontal shift is $h$ units; read the sign carefully.
The vertical changes are applied to the parent $y$ -coordinate.
The point mapping is $(x,y)\mapsto(x+h,ay+k)$ .

Keep each family’s important features

For a linear graph, map two or more points and draw a straight line through them. For a quadratic graph, the parent’s vertex at

(0,0)

moves to

(h,k)

. The vertex is the turning point. The graph opens up when

a>0

and down when

a<0

.

For a square-root graph, the parent endpoint at

(0,0)

moves to

(h,k)

. The endpoint is where the graph begins. The graph extends to the right from that point. If

a<0

, it extends downward instead of upward. The graph’s starting point and direction help you avoid drawing it as a full curve in both directions.

For a reciprocal graph, the parent asymptotes move to

x=h

and

y=k

. Draw these guide lines first, then plot mapped points on each side of the vertical asymptote. When

a>0

, the branch to the right of

x=h

stays above

y=k

, and the branch to the left stays below it. When

a<0

, those positions are reversed. In either case, neither branch reaches an asymptote.

Quadratic vertex: $(h,k)$ ; its opening depends on the sign of $a$ .
Square-root endpoint: $(h,k)$ ; the graph extends rightward.
Reciprocal asymptotes: $x=h$ and $y=k$ .

A dependable sketching routine

First identify the parent function from the equation. Then rewrite the equation, if needed, so the values of

a

,

h

, and

k

are clear. Next choose useful parent points and map them. Plot the new points, mark any vertex, endpoint, or asymptotes, and connect the points using the parent’s shape.

A sketch should match both its points and its features. A quadratic should turn at its vertex; a square-root curve should begin at its endpoint; and reciprocal branches should approach the two asymptotes in the correct regions. If a plotted point seems to contradict one of these features, recheck the parameter signs and the point mapping.

Identify the family before choosing points.
Mark special features before drawing the curve.
Check the mapped points and the curve’s direction against the equation.

Useful parent points and features

Family	Useful points	Feature to preserve
Linear	$(-1,-1),(0,0),(1,1)$	Straight line
Quadratic	$(-2,4),(-1,1),(0,0),(1,1),(2,4)$	Vertex at $(0,0)$
Square-root	$(0,0),(1,1),(4,2)$	Endpoint at $(0,0)$ ; extends right
Reciprocal	$(-2,-\frac12),(-1,-1),(1,1),(2,\frac12)$	Asymptotes $x=0$ and $y=0$

Worked example

Sketch a transformed square-root function

Sketch

y=-2\sqrt{x-1}+3

. Show the endpoint and three additional points.

Identify the parent and parameters
The square root identifies the parent function $f(x)=\sqrt{x}$ . Compare the equation with $y=a\,f(x-h)+k$ . The values are $a=-2$ , $h=1$ , and $k=3$ . The negative value of $a$ reflects the graph across the $x$ -axis, and its magnitude makes a vertical stretch.
$a=-2,\quad h=1,\quad k=3$
Choose parent points
Use parent inputs whose square roots are easy to calculate. The points $(0,0)$ , $(1,1)$ , $(4,2)$ , and $(9,3)$ lie on the parent graph. Mapping these points gives a clear endpoint and shows how the curve extends.
$(0,0),\quad(1,1),\quad(4,2),\quad(9,3)$
Map the points
Add $1$ to each parent $x$ -coordinate. For each $y$ -coordinate, multiply by $-2$ and then add $3$ . The negative multiplier places the curve below its endpoint as it extends to the right.
$(x,y)\mapsto(x+1,-2y+3)$
Plot and connect
The mapped points are $(1,3)$ , $(2,1)$ , $(5,-1)$ , and $(10,-3)$ . Plot them and draw a smooth square-root curve beginning at $(1,3)$ and extending rightward and downward. The endpoint is the highest point.
$(1,3),\quad(2,1),\quad(5,-1),\quad(10,-3)$

Answer: The sketch begins at the endpoint

(1,3)

and curves rightward and downward through

(2,1)

,

(5,-1)

, and

(10,-3)

.

Check: At

x=5

, the equation gives

y=-2\sqrt{4}+3=-1

, matching the mapped point

(5,-1)

.

Worked example

Sketch a transformed reciprocal function

Sketch

y=\frac{2}{x+1}+1

. Show its asymptotes and use four points.

Identify the parent and parameters
The reciprocal expression identifies the parent $f(x)=\frac{1}{x}$ . Rewrite the denominator as $x-(-1)$ . This gives $a=2$ , $h=-1$ , and $k=1$ .
$a=2,\quad h=-1,\quad k=1$
Mark the asymptotes
The parent asymptotes shift according to $h$ and $k$ . Draw the vertical line at $x=-1$ and the horizontal line at $y=1$ as dashed guides before plotting the branches.
$x=-1,\quad y=1$
Choose and map parent points
Use parent points $(1,1)$ , $(2,\frac12)$ , $(-1,-1)$ , and $(-2,-\frac12)$ . Apply the mapping by subtracting $1$ from each $x$ -coordinate and calculating twice the parent $y$ -coordinate plus $1$ .
$(x,y)\mapsto(x-1,2y+1)$
Plot the branches
The mapped points are $(0,3)$ , $(1,2)$ , $(-2,-1)$ , and $(-3,0)$ . The branch right of $x=-1$ lies above $y=1$ ; the branch left of $x=-1$ lies below $y=1$ . Draw each branch approaching the dashed lines without reaching them.
$(0,3),\quad(1,2),\quad(-2,-1),\quad(-3,0)$

Answer: The graph has asymptotes

x=-1

and

y=1

. Its right branch is above the horizontal asymptote, and its left branch is below it.

Check: At

x=0

, the equation gives

y=\frac{2}{1}+1=3

. At

x=-2

, it gives

y=\frac{2}{-1}+1=-1

, confirming two mapped points.

Common mistakes and how to avoid them

Treating

x+4

as a shift right by four units.

Correction: Write

x+4

as

x-(-4)

. Thus

h=-4

, so the shift is four units left.

Adding

h

and

k

to both coordinates of every point.

Correction: Add

h

to the parent

x

-coordinate. Multiply the parent

y

-coordinate by

a

, then add

k

.

Drawing both reciprocal branches on the same side of the horizontal asymptote.

Correction: Use the sign of

a

. If

a>0

, the right branch is above

y=k

and the left branch is below it; if

a<0

, reverse those positions.

Lesson summary

Begin with the correct parent graph and a few useful points.
In $y=a\,f(x-h)+k$ , $h$ shifts horizontally, $k$ shifts vertically, and $a$ stretches, compresses, or reflects.
Map points with $(x,y)\mapsto(x+h,ay+k)$ .
Preserve the family’s shape and mark its vertex, endpoint, or asymptotes.

Check your understanding

Question 1

What is the vertex of

y=(x+2)^2-5

?

$(2,-5)$
$(-2,-5)$
$(-2,5)$
correctIndex: 1

Show answer and explanation

(-2,-5)

Rewrite the bracket as

x-(-2)

. Thus

h=-2

and

k=-5

, so the vertex is

(-2,-5)

.

Question 2

Where does the graph of

y=-\sqrt{x-3}+2

begin?

$(3,2)$
$(-3,2)$
$(3,-2)$
correctIndex: 0

Show answer and explanation

(3,2)

The square-root parent endpoint is

(0,0)

. Here it shifts to

(h,k)=(3,2)

.

Question 3

For

y=\frac{1}{x-4}-2

, what are the asymptotes?

$x=-4$ and $y=2$
$x=4$ and $y=-2$
$x=4$ and $y=2$
correctIndex: 1

Show answer and explanation

x=4

and

y=-2

The denominator gives

h=4

, and the vertical shift gives

k=-2

. The asymptotes are

x=4

and

y=-2

.

Key terms

Parent function: The simplest graph in a function family, used as the starting graph for transformations.
Key point: A point chosen to help locate and sketch a graph.
Vertex: The turning point of a quadratic graph.
Endpoint: The point where a square-root graph begins.
Asymptote: A line that a reciprocal graph approaches but does not reach.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A1.9. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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