Algebra

Conjectures with Functions & Relations

Classify the Function Type

Sort each function into the correct column: Linear, Quadratic, or Exponential.

y = 3x − 7

y = x² + 4x − 5

y = 2ˣ

y = −½x + 3

y = −2x² + 9

y = 5 × 3ˣ

y = (x − 1)(x + 3)

y = 100 × 0.9ˣ

y = 6x

Linear

Quadratic

Exponential

Function Type to General Form

Draw a line from each function type to its general form.

Linear

Quadratic (standard form)

Quadratic (turning point form)

Exponential growth

Exponential decay

y = mx + c

y = ax² + bx + c

y = a(x − h)² + k

y = a × bˣ where b > 1

y = a × bˣ where 0 < b < 1

Linear, Quadratic, or Exponential Table?

Look at the pattern in each table of values. Circle the correct function type.

x: 0, 1, 2, 3, 4 → y: 3, 5, 7, 9, 11 (constant first difference of +2)

Linear

Quadratic

Exponential

x: 0, 1, 2, 3, 4 → y: 1, 4, 9, 16, 25 (first differences: 3, 5, 7, 9 — constant second difference of +2)

Quadratic

Linear

Exponential

x: 0, 1, 2, 3, 4 → y: 2, 6, 18, 54, 162 (each value is ×3)

Exponential

Linear

Quadratic

x: 0, 1, 2, 3, 4 → y: 0, 1, 4, 9, 16 (first differences: 1, 3, 5, 7 — constant second difference of +2)

Quadratic

Linear

Exponential

Identify the y-intercept

Circle the y-intercept (the value of y when x = 0) for each function.

y = 2x + 7

y = x² − 3x + 5

−3

y = 4 × 2ˣ

y = (x − 2)(x + 3)

−6

−2

y = −3x² + 12

−3

Graph Shape to Function Type

Draw a line from each graph description to the function type that produces it.

A straight line with constant slope

A U-shaped curve (parabola) opening upward

A curve that rises slowly then very steeply

An ∩-shaped curve (parabola) opening downward

A curve that starts high and approaches zero

Linear (positive gradient)

Quadratic with a > 0

Exponential growth

Quadratic with a < 0

Exponential decay

Identify the Gradient

For each linear function, circle the gradient (slope).

y = 5x − 3

−3

−5

y = −2x + 8

−2

2y = 6x + 10 (rewrite as y = 3x + 5)

y = ¾x − 1

−1

−¾

Function or Not-a-Function?

Sort each relation: Function or Not a Function. (A function assigns exactly one y-value to each x-value.)

y = 2x + 1

x² + y² = 25 (a circle)

y = x²

x = y² (horizontal parabola)

y = 3ˣ

y = ±√x

y = |x|

x = 4 (vertical line)

Function

Not a Function

Evaluate Functions

Circle the correct value of each function at the given x-value.

f(x) = 2x − 3; find f(4)

−5

g(x) = x² + 1; find g(−3)

−8

h(x) = 3 × 2ˣ; find h(3)

f(x) = (x − 1)(x + 5); find f(0)

−5

−1

g(x) = −x² + 4x; find g(2)

True or False — Quadratic Conjectures

Circle TRUE or FALSE for each conjecture about quadratic functions.

All quadratic functions have exactly two x-intercepts

FALSE

TRUE

The graph of y = ax² + bx + c is always a parabola

TRUE

FALSE

If a > 0 in y = ax² + bx + c, the parabola opens upward

TRUE

FALSE

The vertex of y = x² + bx + c is always on the y-axis

FALSE

TRUE

A quadratic function can have at most one axis of symmetry

TRUE

FALSE

True or False — Exponential Conjectures

Circle TRUE or FALSE for each conjecture about exponential functions.

The graph of y = aˣ (where a > 1) always passes through (0, 1)

TRUE

FALSE

An exponential function can produce negative y-values when a > 0

FALSE

TRUE

The graph of y = 2ˣ has a horizontal asymptote at y = 0

TRUE

FALSE

Exponential decay (0 < b < 1) means y eventually becomes negative

FALSE

TRUE

Multiplying by a constant (e.g. y = 5 × 2ˣ) changes the y-intercept but not the growth rate

TRUE

FALSE

True or False — Linear Conjectures

Circle TRUE or FALSE for each conjecture about linear functions.

A linear function can have a maximum or minimum value

FALSE

TRUE

If two linear functions have the same gradient, they never intersect (assuming different y-intercepts)

TRUE

FALSE

A horizontal line (y = c) is a linear function with gradient 0

TRUE

FALSE

Every straight line can be written in the form y = mx + c

FALSE

TRUE

The gradient of a linear function tells you the rate of change of y with respect to x

TRUE

FALSE

Transformation to Effect on Graph

Draw a line from each transformation to its effect on the graph of y = x².

y = x² + 3

y = (x − 4)²

y = −x²

y = 2x²

y = (x + 1)² − 5

Shift up 3 units

Shift right 4 units

Reflect in the x-axis (flip upside down)

Narrower (steeper) parabola

Shift left 1 unit and down 5 units

Find the Vertex

Circle the correct vertex (turning point) for each quadratic in turning point form y = a(x − h)² + k, where vertex = (h, k).

y = (x − 3)² + 2

(3, 2)

(−3, 2)

(3, −2)

y = −(x + 1)² + 7

(−1, 7)

(1, 7)

(−1, −7)

y = 2(x − 5)² − 4

(5, −4)

(−5, −4)

(5, 4)

y = −3(x + 2)²

(−2, 0)

(2, 0)

(0, −2)

Axis of Symmetry

The axis of symmetry of y = ax² + bx + c is x = −b/(2a). Circle the correct axis of symmetry.

y = x² − 6x + 8 (a = 1, b = −6)

x = 3

x = −3

x = 6

y = 2x² + 8x + 1 (a = 2, b = 8)

x = −2

x = 2

x = −4

y = −x² + 4x − 3 (a = −1, b = 4)

x = 2

x = −2

x = 4

y = 3x² − 12x + 7 (a = 3, b = −12)

x = 2

x = −2

x = 4

Steps to Find the Vertex of y = x² + 4x − 5

Put the steps in the correct order to find the vertex of the parabola y = x² + 4x − 5.

Identify a = 1, b = 4, c = −5

Find the axis of symmetry: x = −b/(2a) = −4/(2×1) = −2

Substitute x = −2 into the equation: y = (−2)² + 4(−2) − 5

Calculate: y = 4 − 8 − 5 = −9

The vertex is (−2, −9)

Since a = 1 > 0, the vertex is a minimum point

Number of x-intercepts from the Discriminant

The discriminant is Δ = b² − 4ac. If Δ > 0: two x-intercepts. If Δ = 0: one x-intercept. If Δ < 0: no x-intercepts. Circle the correct number.

y = x² − 4x + 3 → Δ = (−4)² − 4(1)(3) = 16 − 12 = 4

Two x-intercepts (Δ > 0)

One x-intercept (Δ = 0)

No x-intercepts (Δ < 0)

y = x² + 6x + 9 → Δ = 6² − 4(1)(9) = 36 − 36 = 0

One x-intercept (Δ = 0)

Two x-intercepts (Δ > 0)

No x-intercepts (Δ < 0)

y = 2x² + x + 3 → Δ = 1² − 4(2)(3) = 1 − 24 = −23

No x-intercepts (Δ < 0)

One x-intercept (Δ = 0)

Two x-intercepts (Δ > 0)

y = x² − 5x + 2 → Δ = (−5)² − 4(1)(2) = 25 − 8 = 17

Two x-intercepts (Δ > 0)

One x-intercept (Δ = 0)

No x-intercepts (Δ < 0)

Discriminant Value to Number of Solutions

Sort each discriminant result into the correct column.

Δ = 25

Δ = 0

Δ = −16

Δ = 1

Δ = −3

Two distinct real solutions

One repeated real solution

No real solutions

Parabola Opens Up or Down?

Sort each quadratic function: Opens Upward (a > 0) or Opens Downward (a < 0).

y = x² + 3x + 1

y = −x² + 6x − 8

y = 2x² − 4x + 7

y = −3x² + 2

y = ½x² − x

y = −(x − 4)² + 10

y = 4x² + 1

y = −0.5x² + 3x

Opens Upward (minimum)

Opens Downward (maximum)

Investigate: Coefficient of x² and Parabola Width

Investigate the following conjecture. Show your reasoning with examples.

Conjecture: 'When you increase the coefficient of x² in y = ax², the parabola gets narrower.' Test this by comparing y = x², y = 2x², y = 5x², and y = ½x². For each, calculate y when x = 1, 2, and 3. What happens to the y-values as 'a' increases? Explain why a larger 'a' makes the parabola narrower.

Investigate: When Does Exponential Overtake Linear?

Test the conjecture and show your working.

Conjecture: 'An exponential function will always eventually exceed a linear function.' Test with y = 2ˣ and y = 100x. Calculate both functions for x = 0, 1, 2, 5, 7, 8, 10, and 15. For what values of x is the linear function greater? At approximately what x-value does the exponential overtake? Explain why this must always happen eventually.

Find and Verify x-intercepts

Find the x-intercepts of each quadratic and verify your answer.

Find the x-intercepts of y = x² − 2x − 15 by factorising. Then verify by substituting each x-intercept back into the equation. Also find the vertex and sketch a rough graph labelling the intercepts and vertex.

Find the x-intercepts of y = 2x² + 5x − 3 by factorising or using the quadratic formula. Verify your solutions. How does the discriminant confirm the number of intercepts?

Compare Two Functions for Same Output

Find where two functions produce the same y-value.

For what value(s) of x do f(x) = x² − 3 and g(x) = 2x + 1 give the same output? Set them equal and solve: x² − 3 = 2x + 1. Verify your solutions by substituting into both functions. What does this represent graphically?

A linear function f(x) = 50 + 10x and a quadratic function g(x) = x² model two different situations. Find where they intersect. For what x-values is the linear function greater? When does the quadratic overtake?

Investigate Translations

Explore how translations affect a function's graph.

Starting with y = x², investigate what happens when you replace x with (x − h). Compare y = x², y = (x − 3)², and y = (x + 2)². For each, find the vertex and y-intercept. State your conjecture about how the value of h affects the graph's position.

Starting with y = x², investigate what happens when you add k. Compare y = x², y = x² + 4, and y = x² − 3. For each, find the vertex. Then combine both: describe the graph of y = (x − 1)² + 6 without plotting it.

True or False — Domain and Range

Circle TRUE or FALSE for each statement about function domains and ranges.

The domain of y = x² is all real numbers

TRUE

FALSE

The range of y = x² is y ≥ 0

TRUE

FALSE

The domain of y = 2ˣ is x ≥ 0 only

FALSE

TRUE

The range of y = −x² + 5 is y ≤ 5

TRUE

FALSE

The range of y = 3ˣ includes negative numbers

FALSE

TRUE

The range of a linear function y = mx + c (where m ≠ 0) is all real numbers

TRUE

FALSE

Create a Conjecture and Test It

Make your own conjecture about functions and test it systematically.

Create a conjecture about the relationship between the sign of 'a' and the direction a parabola opens, AND the number of x-intercepts based on the vertex's position. For example: 'If a > 0 and the vertex is below the x-axis, then...' State your conjecture clearly, test it with at least 3 specific quadratics, and explain whether your conjecture holds or needs revision.

Real-World Function Modelling

Apply function knowledge to a real-world scenario.

A cricket ball is hit into the air. Its height (in metres) after t seconds is modelled by h(t) = −5t² + 20t + 1.5. Find: (a) the initial height when hit, (b) the maximum height and when it occurs, (c) when the ball returns to ground level (h = 0). Explain what each part of the equation represents in context.

A company's profit (in thousands of dollars) is modelled by P(x) = −2x² + 24x − 40, where x is the price of the product in dollars. Find the price that maximises profit, the maximum profit, and the price range where the company makes a positive profit. What does the vertex represent in this context?

Graphing with Technology

Use a graphing tool (such as Desmos, GeoGebra, or a graphing calculator) to explore functions.

1Graph y = ax² for a = −3, −1, −0.5, 0.5, 1, 3. Make a conjecture about how the value of 'a' affects the parabola's shape and direction. Test with one more value of 'a' to verify.
2Graph y = (x − h)² + k for different values of h and k. Explain how to move a parabola to any position on the coordinate plane.
3Graph y = 2ˣ, y = 3ˣ, and y = 10ˣ on the same axes. Also graph y = 0.5ˣ and y = 0.1ˣ. Make a conjecture about how the base affects the steepness and whether the function grows or decays.

Real-World Data and Function Fitting

Collect or research real data and decide which type of function best models it.

1Collect data on the temperature at different times of day (e.g. 6 am, 9 am, 12 pm, 3 pm, 6 pm, 9 pm). Plot the points. Does the pattern look linear, quadratic, or something else? Justify your choice.
2Research Australia's population over the last 50 years (e.g. every 10 years). Plot the data. Test whether a linear or exponential model fits better by comparing predictions.
3Drop a ball from different heights and measure the bounce height. Plot drop height vs bounce height. Is the relationship linear? Find an approximate equation for the line of best fit.

Quadratic Functions — Key Features

Identify and explain all key features of a quadratic function.

For f(x) = x² − 6x + 8: (a) Find x-intercepts by factoring. (b) Find y-intercept. (c) Find the axis of symmetry. (d) Find the vertex. (e) State the minimum value.

For g(x) = −2x² + 8x − 6: (a) State whether parabola opens up or down. (b) Find axis of symmetry. (c) Find vertex. (d) Find x-intercepts. (e) State maximum value.

Function Vocabulary

Draw a line from each function term to its correct definition.

Domain

Range

x-intercept

y-intercept

Axis of symmetry

The value of y when x = 0

The vertical line through the vertex of a parabola

The set of all possible output (y) values

The value(s) of x when y = 0

The set of all allowable input (x) values

Parabola Features — Identify from Equation

Circle the correct answer for each feature.

f(x) = 3(x − 2)² + 5 — vertex is at:

(2, 5)

(−2, 5)

(2, −5)

f(x) = −(x + 3)² + 7 — the parabola:

Opens downward, vertex at (−3, 7)

Opens upward, vertex at (3, 7)

Opens downward, vertex at (3, −7)

f(x) = 2x² − 4x + 1 — axis of symmetry is:

x = 1

x = 2

x = −1

f(x) = x² + 6x + 9 — discriminant is:

0 (one repeated root)

Positive (two roots)

Negative (no real roots)

Function vs Not a Function

Sort each relation: Function or Not a Function.

y = x² (each x gives one y)

x = y² (two y values for each x > 0)

y = 3x + 1 (straight line)

A circle: x² + y² = 25

y = |x| (absolute value)

y = sin(x)

{(1,2), (2,3), (1,4)} (x = 1 maps to two outputs)

y = √x (for x ≥ 0)

Function

Not a Function

The Discriminant — Nature of Roots

Use the discriminant to determine the nature of roots without solving.

For each quadratic, calculate b² − 4ac and state whether there are 0, 1, or 2 real roots: (a) x² + 5x + 6 = 0 (b) x² + 4x + 4 = 0 (c) x² + x + 3 = 0 (d) 2x² − 3x − 5 = 0

Steps for Solving a Quadratic by Factoring

Put the steps in the correct order.

Write the equation in standard form: ax² + bx + c = 0

Look for factors: find two numbers that multiply to ac and add to b

Rewrite the middle term using those factors

Factor by grouping or use the factored form

Set each factor equal to zero

Solve each resulting linear equation

Check both solutions in the original equation

Quadratic Formula

Apply the quadratic formula to solve equations that do not factorise easily.

Solve using the quadratic formula: 2x² − 5x − 3 = 0. State the values of a, b, c. Calculate the discriminant. Find both solutions. Verify by substitution.

Solve: x² + 3x − 1 = 0. Express solutions in surd form and as decimals correct to 2 decimal places.

Quadratic Roots to Equation Form

Draw a line from each pair of roots to the factored form of the quadratic.

x = 3 and x = −2

x = 0 and x = 5

x = −4 and x = −1

x = 1 (repeated)

x = ½ and x = −3

(x − ½)(x + 3) = 0

(x − 1)² = 0

(x − 3)(x + 2) = 0

x(x − 5) = 0

(x + 4)(x + 1) = 0

Completing the Square

Use completing the square to rewrite quadratics in vertex form.

Rewrite f(x) = x² − 8x + 5 in the form (x − h)² + k by completing the square. State the vertex and axis of symmetry.

Rewrite f(x) = 2x² + 12x + 7 in vertex form. (Hint: first factor out 2 from the x² and x terms.) State the vertex.

Transformations of Parabolas

Circle the correct description of each transformation.

y = x² shifted to y = x² + 3 means:

Vertical shift up 3 units

Vertical shift down 3 units

Horizontal shift right 3 units

y = x² shifted to y = (x − 4)² means:

Horizontal shift right 4 units

Horizontal shift left 4 units

Vertical shift up 4 units

y = x² changed to y = −x² means:

Reflection in the x-axis (flips upside down)

Reflection in the y-axis

Vertical stretch

y = x² changed to y = 3x² means:

Vertical stretch by factor 3 (narrower)

Horizontal stretch (wider)

Shift up 3

Conjectures in Algebra — Testing and Proving

Formulate, test, and attempt to prove algebraic conjectures.

Conjecture: 'The sum of any two consecutive integers is always odd.' Test with 3 examples. Then prove it algebraically using n and n+1.

Conjecture: 'The product of any two consecutive even integers is divisible by 8.' Test with 3 examples. Prove algebraically using 2n and 2(n+1).

Functions in Context — Projectile Motion

Apply quadratic functions to model projectile motion.

A ball is thrown upward from a height of 2 m with height h(t) = −5t² + 20t + 2 (h in metres, t in seconds). (a) What is the initial height? (b) Find the maximum height and when it occurs. (c) When does the ball hit the ground? (Solve h(t) = 0) (d) What is the height after 3 seconds?

Classify the Relationship — Function Type

Sort each context into the type of function that models it best.

Height of a ball thrown upward over time

Saving $50 per week from zero

Compound interest on a savings account

Area of a square room as its side length increases

Temperature dropping by equal amounts per hour

Bacterial population doubling every hour

Distance fallen by an object under gravity (d = ½gt²)

Cost of a taxi: $3 flag-fall + $2.50/km

Linear (y = mx + b)

Quadratic (y = ax² + bx + c)

Exponential (y = abˣ)

Quadratic Functions — Real World

Find quadratic functions in real-world settings.

1Throw a ball straight up and catch it. Measure the time in the air. Using h = −5t² + vt, estimate the initial velocity v. What is the maximum height reached?
2Research the path of a basketball free throw. Estimate the launch angle and initial speed. Sketch the parabolic path and identify the vertex.
3Find a bridge arch or architectural arch. Measure or estimate its dimensions and write a quadratic equation that models the arch.

Polynomial Degree and End Behaviour

Analyse the end behaviour of polynomial functions.

For f(x) = x³ − 3x² + 2x: (a) What is the degree? (b) As x → +∞, what does f(x) do? (c) As x → −∞, what does f(x) do? (d) Find x-intercepts by factoring.

Compare the end behaviour of y = x² and y = x³. Explain why even-degree polynomials have 'matching' ends while odd-degree polynomials have 'opposite' ends.

Types of Quadratic Solutions

Tally the type of solution found for each quadratic in a set of 25 problems.

Item	Tally	Total
Two distinct real roots
One repeated root
No real roots (complex)

Inverse Functions — Introduction

Explore the concept of inverse functions.

For f(x) = 2x + 3: (a) Find f⁻¹(x) by swapping x and y and solving for y. (b) Verify that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. (c) What is the graphical relationship between f and f⁻¹?

Explain why f(x) = x² (for all real x) does not have an inverse function, but f(x) = x² for x ≥ 0 does. What is the inverse of the restricted version?

Identify the Function from Its Graph Description

Circle the function type that matches each description.

A U-shaped curve with vertex at the origin, symmetrical about the y-axis

Quadratic: y = x²

Linear: y = x

Absolute value: y = |x|

A curve that passes through (0,1) and increases very steeply as x increases

Exponential: y = 2ˣ

Quadratic: y = x²

Linear: y = 2x

A straight line with negative gradient, crossing both axes

Linear: y = −2x + 4

Quadratic: y = −x²

Constant: y = 4

A curve shaped like an upside-down U, with a maximum turning point

Quadratic: y = −x² + 4

Exponential decay

Linear with negative slope

Conjecture About Even and Odd Functions

Investigate symmetry properties of functions.

Define what it means for a function to be 'even'. Give two examples.

Define what it means for a function to be 'odd'. Give two examples.

Is f(x) = x² + 3 even, odd, or neither? Justify using algebra.

Is f(x) = x³ − x even, odd, or neither? Show working.

Match Function to Its Graph Feature

Draw a line from each function to the graph feature that best describes it.

y = x²

y = 2ˣ

y = 1/x

y = √x

y = |x|

y = x³

V-shaped, vertex at origin

Passes through (0,1), always positive

Parabola opening upward

Defined only for x ≥ 0

Rotational symmetry about origin

Hyperbola with two branches

Classify Functions by Type

Sort each function into the correct category: Linear, Quadratic, Exponential, or Other.

y = 3x − 7

y = x² + 2x − 1

y = 5 × 2ˣ

y = 1/x

y = −4x + 9

y = (x + 3)²

y = 0.5ˣ

y = √(x + 1)

Linear

Quadratic

Exponential

Other

Investigating Translations of Parabolas

Explore how changing constants shifts the graph of y = x².

Describe how the graph of y = x² + k differs from y = x² for positive k.

Describe how y = (x − h)² differs from y = x².

Write the equation of a parabola with vertex at (3, −2) and opening upward.

Sketch the graph of y = (x − 1)² + 4 and label the vertex and axis of symmetry.

Draw here

Function Values — Continue the Pattern

Complete each table of function values.

Domain and Range Exploration

Determine the domain and range for each function.

State the domain and range of f(x) = x². Is any real number excluded?

State the domain and range of f(x) = √x. Why is the domain restricted?

State the domain and range of f(x) = 1/x. Which value of x is excluded and why?

Create a function whose domain is all real numbers but whose range is only positive numbers.

Types of Functions Encountered

Tally how many of each function type you have studied this week.

Item	Tally	Total
Linear
Quadratic
Exponential
Hyperbolic
Square root

Relations vs Functions

Determine whether each relation is also a function and justify your answer.

Is {(1,2),(2,3),(3,4)} a function? Explain using the definition.

Is {(1,2),(1,3),(2,4)} a function? What rule does it violate?

Explain the vertical line test and why it works.

Give an example of a relation that is NOT a function. Draw its graph.

Draw here

Function Notation — Evaluate Correctly

Circle the correct value of the function.

If f(x) = 2x + 3, then f(4) =

If g(x) = x² − 1, then g(−3) =

−8

−10

If h(x) = 3x − 5, then h(0) =

−5

−3

If f(x) = x² + x, then f(2) =

Composite and Inverse Functions

Explore how functions can be combined and reversed.

If f(x) = 2x and g(x) = x + 3, find f(g(x)) and g(f(x)).

Are f(g(x)) and g(f(x)) the same? What does this tell you about composition?

Find the inverse of f(x) = 3x − 6. Show all steps.

Check your inverse by computing f(f⁻¹(x)). What should you get?

Functions in Everyday Technology

Explore where mathematical functions appear in technology around you.

1Find a graph on your phone's screen time or health app. Identify the type of function it resembles.
2Look at a population graph for Australia. Is the growth linear, quadratic, or exponential? Justify.
3Open a spreadsheet and enter the formula =A1^2 for x = 1..10. Plot the result and describe the shape.
4Research how GPS uses mathematical functions to calculate your position. Write a short paragraph.
5Ask a family member what functions or graphs they use in their job. Report back to your class.

Piecewise Functions

Interpret and create piecewise-defined functions.

A taxi charges $3 flag fall plus $2/km. Write a function for the fare for x km.

A phone plan charges $0.10/min for the first 100 minutes, then $0.05/min after. Write a piecewise function for cost.

Evaluate your phone plan function at 80 minutes and at 150 minutes.

Draw a graph of the phone plan function for 0 ≤ x ≤ 200.

Draw here

Transformations of Exponential Functions

Investigate how changing parameters affects exponential graphs.

Describe how y = 2^x + 3 differs from y = 2^x. What is the y-intercept of each?

Describe how y = 2^(x−1) differs from y = 2^x. In which direction is it shifted?

Sketch both y = 2^x and y = −2^x on the same axes. What is the relationship between them?

Draw here

Write the equation of an exponential function that has been reflected in the x-axis and shifted up by 4 units. State its horizontal asymptote.

Function Types Graphed This Week

Tally each function type you graphed during this unit.

Item	Tally	Total
Linear
Quadratic
Exponential
Hyperbolic
Piecewise

Sketching Quadratic Functions

Sketch parabolas by identifying key features.

For y = x² − 4x + 3, find the x-intercepts, y-intercept, axis of symmetry, and vertex. Show all working.

Sketch the parabola and label all key features found above.

Draw here

Convert y = x² − 4x + 3 to vertex form y = (x−h)² + k. Show the completing-the-square process.

What to do next

Measurement

Surface Area & Volume of Composites

Solve problems involving surface area and volume of composite objects by decomposing them into known shapes