Algebra

Graphing Quadratic Functions

Key Features Vocabulary

Draw a line from each term to its correct description.

Vertex

Axis of symmetry

x-intercept(s)

y-intercept

Parabola

The turning point -- highest or lowest point on the parabola

The vertical line passing through the vertex

The point(s) where the parabola crosses the x-axis

The value of y when x = 0

The U-shaped curve formed by a quadratic function

Table of Values for y = x^2 - 4

Complete the table of values for y = x^2 - 4.

x = -3: y = ___ x = -2: y = ___ x = -1: y = ___ x = 0: y = ___ x = 1: y = ___ x = 2: y = ___ x = 3: y = ___

Identify Key Features from the Rule

For each parabola, state the y-intercept, x-intercepts and vertex.

y = x^2 - 4: Y-intercept: ___ X-intercepts: ___ Vertex: ___

y = x^2 - 6x + 8: Factorise: ___ X-intercepts: ___ Axis of symmetry: x = ___ Vertex: ___

y = x^2 + 2x - 3: Factorise: ___ X-intercepts: ___ Axis of symmetry: x = ___ Vertex: ___

Features of y = x^2

Circle the correct answer for each question about the basic parabola y = x^2.

The vertex of y = x^2 is at:

(0, 0)

(1, 0)

(0, 1)

The parabola y = x^2 opens:

Upward

Downward

Sideways

The axis of symmetry of y = x^2 is:

x = 0

y = 0

x = 1

For y = -x^2, the parabola:

Opens downward

Opens upward

Is a straight line

Effects of Parameter Variation

Compare each parabola to y = x^2 and describe how it has changed.

y = 2x^2:

y = x^2 + 3:

y = (x - 4)^2:

y = -x^2:

Sketching Parabolas

Sketch each parabola. Label the vertex, axis of symmetry and intercepts.

y = x^2 - 9 (Hint: x-intercepts at x = +/-3, vertex at (0, -9))

Draw here

y = x^2 + 4x + 3 (Hint: factorise first)

Draw here

Quadratic Word Problem

Use a quadratic function to solve this problem.

A ball is thrown upward and its height (in metres) after t seconds is h = -5t^2 + 20t. Find the maximum height and the time when the ball hits the ground. Show all working.

Identifying Vertex and Axis of Symmetry from a Graph

Read the key features from each parabola description and record them.

A parabola passes through (-4, 0) and (2, 0) and has its vertex between them. State the axis of symmetry (x = ___) and find the vertex x-coordinate.

A parabola has axis of symmetry x = 3 and passes through (3, -5). State the vertex:

For y = x^2 - 8x + 7, find the x-intercepts by factorising, then find the axis of symmetry and vertex.

Effect of Changing Coefficient a

Circle the correct description for each change to the coefficient a in y = ax^2.

Compared to y = x^2, the parabola y = 3x^2 is:

Narrower than y = x^2

Wider than y = x^2

The same width

Compared to y = x^2, the parabola y = 0.5x^2 is:

Narrower than y = x^2

Wider than y = x^2

Reflected

Compared to y = x^2, the parabola y = -2x^2:

Opens upward, narrower

Opens downward, narrower

Opens upward, wider

A larger value of a (e.g. a = 5 vs a = 1) makes the parabola:

Narrower

Wider

Shift up

Effect of Changing c -- Vertical Shift

Compare each parabola to y = x^2 and describe the transformation.

y = x^2 + 5 compared to y = x^2: vertex moves from (0, 0) to ___. The parabola shifts ___.

y = x^2 - 3 compared to y = x^2: vertex moves to ___. The parabola shifts ___.

Write a rule: changing c in y = x^2 + c causes the parabola to:

Sketching Parabolas from Tables of Values

Complete the table of values and sketch the parabola. Label the vertex and axis of symmetry.

y = x^2 + 2x - 8 x = -4: y = ___ x = -3: y = ___ x = -1: y = ___ x = 0: y = ___ x = 1: y = ___ x = 2: y = ___ Sketch and label vertex:

Draw here

Match Parabola to Its Features

Sort each equation into the correct column based on whether its vertex is at the origin, above the x-axis, or below the x-axis.

y = x^2

y = x^2 + 4

y = x^2 - 9

y = -x^2

y = x^2 + 1

y = x^2 - 5

Vertex at origin (0, 0)

Vertex above x-axis

Vertex below x-axis

Parabolas Around the Home

Spot and investigate parabolic shapes in everyday life.

1Throw a ball across the room and photograph its path. Sketch the parabola and estimate where the vertex (highest point) is.
2Research how satellite dish shapes use parabolas. Draw a diagram showing how a signal reflects off the dish to a single focal point.
3Using graphing software or a graphics calculator, plot y = x^2, y = 2x^2 and y = 0.5x^2 on the same axes. Write two observations about how a affects the shape.

Finding the Vertex

Find the vertex of each parabola by completing the axis of symmetry step.

Find the vertex of y = x² − 4x + 7. Show full working.

Find the vertex of y = −x² + 6x − 5. Show full working.

Find the vertex of y = 2x² − 4x + 1. Show full working.

TipFind x = −b/(2a), then substitute into the equation to find the y-coordinate of the vertex.

Match Equation to Vertex

Match each quadratic equation to its vertex.

y = x² − 2x + 1

y = x² + 4x + 4

y = x² − 6x + 9

y = −x² + 2x − 1

Vertex (1, 0)

Vertex (3, 0)

Vertex (−2, 0)

Vertex (1, 0) — opens down

TipUse x = −b/(2a) and substitute back to find the vertex.

Identify the Vertex

Circle the correct vertex for each parabola.

y = x² − 2x − 3

(1, −4)

(0, −3)

(2, −3)

(−1, 0)

y = x² + 4x + 3

(−2, −1)

(0, 3)

(2, 15)

(−4, 3)

y = −x² + 4x

(2, 4)

(0, 0)

(4, 0)

(−2, −12)

Finding X-intercepts

Find the x-intercepts of each quadratic by factorising.

Find the x-intercepts of y = x² − 5x + 6.

Find the x-intercepts of y = x² + x − 6.

Find the x-intercepts of y = x² − 9.

Find the x-intercepts of y = x(x − 4).

TipSet y = 0, factorise, then solve each bracket equal to zero.

Sketch a Parabola — Five Key Points

Sketch each parabola using its five key features.

Sketch y = x² − 4x + 3. Label the vertex, axis of symmetry, and all intercepts.

Draw here

List the five key features you used: vertex, axis, y-intercept, and x-intercepts.

TipFind: (1) y-intercept, (2) axis of symmetry, (3) vertex, (4) x-intercepts, (5) a symmetric point.

Vertex Form — Read the Vertex

Match each vertex form equation to its vertex.

y = (x − 3)² + 2

y = (x + 1)² − 4

y = −(x − 2)² + 5

y = 2(x + 3)² + 1

Vertex (3, 2)

Vertex (−1, −4)

Vertex (2, 5)

Vertex (−3, 1)

Converting to Vertex Form

Complete the square to convert each equation to vertex form.

Convert y = x² − 6x + 11 to vertex form. Show full working.

Convert y = x² + 4x + 1 to vertex form. Show full working.

State the vertex and axis of symmetry for each equation above.

TipStep 1: group the x-terms; Step 2: complete the square; Step 3: balance by adding/subtracting the same value.

Sort by Parabola Width

Sort these parabolas from widest to narrowest.

y = 0.1x²

y = 0.5x²

y = x²

y = 2x²

y = 5x²

y = 10x²

Widest

Medium

Narrowest

Translations of y = x²

Circle the correct description of each transformation.

y = x² + 3 compared to y = x²

Shift 3 up

Shift 3 down

Shift 3 right

Shift 3 left

y = x² − 5 compared to y = x²

Shift 5 up

Shift 5 down

Shift 5 right

Shift 5 left

y = (x − 2)² compared to y = x²

Shift 2 right

Shift 2 left

Shift 2 up

Shift 2 down

y = (x + 4)² compared to y = x²

Shift 4 right

Shift 4 left

Shift 4 up

Shift 4 down

Sketching from Vertex Form

Sketch each parabola given in vertex form.

Sketch y = (x − 1)² − 4. Label vertex, axis of symmetry, and x-intercepts.

Draw here

What are the x-intercepts of y = (x − 1)² − 4? Show your working.

Sketch y = −(x + 2)² + 9. Label all key features.

Draw here

TipPlot the vertex, then find 2 more points symmetrically on each side.

Parabola in Context — Projectile

Model a projectile with a quadratic equation.

The height of a ball (in metres) after t seconds is h = −5t² + 20t. Find the maximum height and when it occurs.

When does the ball return to the ground? Show your working.

Sketch the path of the ball. Label the vertex and the two time-intercepts.

Draw here

TipMaximum height is the y-coordinate of the vertex. The ball hits the ground when y = 0.

Match Description to Equation

Match each description to the correct quadratic equation.

Vertex (0,0), opens up

Vertex (0, 3), opens up

Vertex (2, 0), opens up

Vertex (2, 3), opens down

y = x²

y = x² + 3

y = (x − 2)²

y = −(x − 2)² + 3

Comparing Two Parabolas

Analyse and compare two quadratic functions.

Compare y = x² − 4x + 3 and y = −x² + 4x − 3. What do you notice about their equations?

Find the vertex and intercepts of each. How are the two parabolas geometrically related?

Sketch both on the same axes and describe the relationship.

Draw here

TipCompare direction, width, vertex, intercepts, and maximum or minimum values.

Maximum or Minimum

Circle whether each parabola has a maximum or minimum.

y = x² + 3x − 5

Minimum

Maximum

y = −2x² + 4x + 1

Minimum

Maximum

y = 5x² − x

Minimum

Maximum

y = −x² − x − 1

Minimum

Maximum

Revenue and Profit — Quadratic Application

Use quadratic functions to model business problems.

A café sells coffee at $4 per cup and sells 200 cups per day. For every $1 increase in price, they sell 20 fewer cups. Write a revenue function R(x) where x is the price increase.

Find the price increase that maximises revenue. What is the maximum revenue?

What price per cup maximises revenue?

TipRevenue = price × quantity. If quantity decreases as price increases, the product is a quadratic.

Investigating Quadratics with Technology

Use Desmos or a graphing calculator to explore quadratic functions.

On Desmos, plot y = a·x² with a slider for a. Describe how a affects (i) direction of opening, (ii) width of the parabola.

Plot y = x² + k with a slider for k. Describe how k affects the graph.

Plot y = (x − h)² with a slider for h. Describe how h affects the graph.

Write the equation of a parabola that passes through (0, 0) and has vertex at (3, 9). Verify using Desmos.

TipTechnology allows you to test conjectures quickly — use it to investigate, not to avoid thinking.

Match Form to Its Advantage

Match each quadratic form to its main advantage.

y = ax² + bx + c

y = a(x − p)(x − q)

y = a(x − h)² + k

A sketch or graph

Immediately shows x-intercepts

Immediately shows vertex

Shows all features visually

Shows y-intercept directly from c

Choosing the Right Form

Decide which form is most useful for each task.

You need to find the x-intercepts of a quadratic. Which form would you use? Why?

You need to find the maximum height of a thrown ball. Which form would you use? Why?

You need to find where a parabola crosses the y-axis. Which form would you use? Why?

TipUsing the right form saves time — choose strategically.

Design a Quadratic Situation

Create your own real-world quadratic scenario.

Describe a real-world situation that could be modelled by a quadratic function. Be specific about what each variable represents.

Write a quadratic equation for your situation. State the vertex and what it means in context.

Sketch the graph and label all key features with real-world interpretations.

Draw here

TipReal-world quadratics appear in physics, business, engineering, and sport.

Parabola Hunt

Investigate quadratic functions beyond the worksheet.

1Open Desmos and create a 'parabola art' picture using at least 5 different quadratic equations with restricted domains. Share your creation.
2Research the height formula for a projectile on the Moon (g ≈ 1.6 m/s²) versus Earth (g ≈ 9.8 m/s²). If you jump upward at 3 m/s, how much higher would you go on the Moon? Show calculations.
3Find a photo of a bridge, arch, or satellite dish. Sketch the parabolic cross-section and estimate an equation using landmarks in the photo as coordinate guides.

Quadratic Functions — Reflection

Reflect on your learning about quadratic functions.

List the five key features of a parabola and explain how you find each one from the equation.

Which form of the quadratic (standard, factored, or vertex) do you find easiest to work with? Why?

Describe one real-world application of quadratic functions that interests you.

What aspect of graphing parabolas do you still find challenging? How could you practise it?

TipMetacognitive reflection — thinking about your thinking — is one of the most powerful study strategies.

Parabola Problem Solving

Solve each multi-step quadratic problem.

A stone is dropped from a cliff. Its height in metres after t seconds is h = 100 − 5t². When does it hit the ground? What is the height after 3 seconds?

A garden bed is x metres wide. Its length is (10 − x) metres. Write the area as a quadratic. What width maximises the area?

Which Equation Matches the Description?

Circle the equation that matches each description.

A parabola with vertex at (2, −3) opening upward

y = (x − 2)² − 3

y = (x + 2)² − 3

y = −(x − 2)² − 3

y = (x − 2)² + 3

A parabola with x-intercepts at x = −1 and x = 4

y = (x + 1)(x − 4)

y = (x − 1)(x + 4)

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

y = (x − 1)(x − 4)

A parabola with y-intercept at (0, 6) and vertex above the x-axis opening downward

y = −x² + 6

y = x² + 6

y = (x − 6)²

y = −(x + 6)²

Writing a Quadratic from a Graph

Write a quadratic equation from the given key features.

A parabola has x-intercepts at x = 1 and x = 5, and passes through (3, −4). Write its equation.

A parabola has vertex at (2, 8) and passes through (0, 0). Write its equation.

TipUse y = a(x − p)(x − q) if you know x-intercepts, or y = a(x − h)² + k if you know the vertex.

Quadratic Application Contexts

Match each real-world situation to the relevant quadratic feature.

Maximum height of a ball

When a ball hits the ground

When projectile is at launch level

Starting height of projectile

Y-intercept

X-intercepts

Vertex y-coordinate

X-intercepts (both values)

Year 9 Quadratics — Comprehensive Problem

Solve a complete quadratic problem from start to finish.

For y = x² − 2x − 8: (a) Find the y-intercept.

(b) Factorise and find the x-intercepts.

(d) Find the vertex coordinates.

(e) Sketch the parabola, labelling all features found above.

Draw here

TipWork systematically: find all five key features before sketching.

Quadratic Functions Investigation

Deepen your understanding of quadratic functions through exploration.

1Research the equation of a suspension bridge cable (it is approximately a parabola). Find a famous suspension bridge and estimate its quadratic equation from a photograph using a coordinate grid overlay.
2Use a ball and a ruler to estimate the equation of a parabolic throw. Film in slow motion, then plot the trajectory from 5 data points. Use technology to find the best-fit quadratic.
3Investigate the relationship between the vertex form and factored form of a quadratic using the equation y = a(x − r₁)(x − r₂) where r₁ and r₂ are x-intercepts. Derive the vertex x-coordinate algebraically.

Transformations of y = x²

Describe the transformation applied to y = x² to produce each graph.

Describe the transformation from y = x² to y = (x − 4)² + 3.

Describe the transformation from y = x² to y = −3x².

Write the equation of a parabola that is y = x² shifted 2 left and 5 down.

Write the equation of y = x² reflected in the x-axis and stretched vertically by factor 4.

TipThink of each change to the equation as a geometric transformation: shift, reflect, or stretch.

Steps to Sketch a Parabola

Order the steps for sketching y = x² + bx + c.

Identify a, b, c from the equation

Calculate the axis of symmetry x = −b/(2a)

Substitute to find the vertex y-coordinate

Substitute x = 0 to find the y-intercept

Factorise or use the quadratic formula to find x-intercepts

Plot all key points and draw smooth curve

TipA logical order makes the sketch accurate and complete.

Graphing Quadratics — Synthesising Skills

Use all your graphing skills to completely analyse each quadratic.

For y = 2x² − 8x + 6, find: (a) y-intercept, (b) axis of symmetry, (c) vertex, (d) x-intercepts by factorising, (e) sketch.

For y = −x² + 2x + 3, find all key features and sketch.

TipThis is an exam-level exercise — be systematic and show all working.

Parabola Symmetry Applications

Use the axis of symmetry to find the missing value.

A parabola has axis x = 3. If one x-intercept is at x = 1, where is the other?

x = 5

x = 6

x = 4

x = 2

A parabola has axis x = −2. If one x-intercept is at x = 0, where is the other?

x = −4

x = 2

x = −2

x = 4

A parabola has vertex at (4, −3). If the point (2, 5) is on the parabola, another point is:

(6, 5)

(6, −5)

(2, −5)

(0, 5)

Quadratic Modelling — Extended Investigation

Model a real situation with a quadratic function.

A farmer has 60 m of fencing and wants to fence a rectangular paddock along a river (one side is the river, so only 3 sides need fencing). Let x = width perpendicular to river. Write Area A as a quadratic function of x.

Find the value of x that maximises area. What are the dimensions of the optimal rectangle?

What is the maximum area? Compare this to a square paddock using the same 60 m of fencing.

TipGood mathematical modelling involves setting up the equation, finding key features, and interpreting results in context.

Parabola — Peer Teaching Task

Prepare a mini-lesson on graphing parabolas.

Write a step-by-step guide for graphing y = x² + bx + c that a student who has never seen parabolas could follow.

Include a worked example in your guide.

Write one practice question for your guide, with a full worked solution.

TipThe best way to consolidate understanding is to explain it clearly to someone else.

Quadratics — Final Vocabulary Match

Match each term to its correct definition.

Parabola

Vertex

Axis of symmetry

Discriminant

Y-intercept

The value b² − 4ac, determines number of x-intercepts

The graph of any quadratic function

The turning point (maximum or minimum) of a parabola

The point where the parabola crosses the y-axis

The vertical line that mirrors the parabola

Graphing Quadratics — Final Portfolio Task

Demonstrate mastery of graphing quadratic functions.

Sketch three parabolas on the same axes: y = x², y = 2x², and y = 0.5x². Describe the effect of changing a.

Draw here

Sketch three parabolas on the same axes: y = x², y = x² + 3, and y = x² − 3. Describe the effect of adding a constant.

Draw here

Write a general statement: how does each of the parameters a, h, and k in y = a(x − h)² + k affect the graph of y = x²?

TipThis is a comprehensive task — plan your response before you write.

Graph and Equation Matching

Circle the equation that best matches each graph description.

Opens upward, vertex at (0, −4), two x-intercepts

y = x² − 4

y = −x² − 4

y = (x − 4)²

y = x² + 4

Opens downward, vertex at (1, 9), crosses x-axis at two points

y = −(x − 1)² + 9

y = (x − 1)² + 9

y = −(x + 1)² + 9

y = −(x − 1)² − 9

Opens upward, vertex at (3, 0), only one x-intercept

y = (x − 3)²

y = x² − 3

y = (x + 3)²

y = x² + 3

100

Self-Assessment: Graphing Quadratic Functions

Assess your own understanding and identify next steps.

Rate your confidence (1–5) with: finding the vertex, finding x-intercepts, sketching parabolas. Explain your ratings.

Write one question about quadratic functions that you can now answer that you could not answer before this worksheet.

What is one thing about quadratic functions you would like to explore further?

TipHonest self-assessment is the starting point for targeted improvement.

What to do next

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