lesson 3 3 problem solving writing functions answer key

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Unit 14: Quadratic functions & equations

About this unit.

We've seen linear and exponential functions, and now we're ready for quadratic functions. We'll explore how these functions and the parabolas they produce can be used to solve real-world problems.

Intro to parabolas

• Parabolas intro (Opens a modal)
• Interpreting a parabola in context (Opens a modal)
• Interpret a quadratic graph (Opens a modal)
• Parabolas intro Get 3 of 4 questions to level up!
• Interpret parabolas in context Get 3 of 4 questions to level up!
• Interpret a quadratic graph Get 3 of 4 questions to level up!

Solving and graphing with factored form

• Zero product property (Opens a modal)
• Graphing quadratics in factored form (Opens a modal)
• Quadratic word problems (factored form) (Opens a modal)
• Zero product property Get 3 of 4 questions to level up!
• Graph quadratics in factored form Get 3 of 4 questions to level up!
• Quadratic word problems (factored form) Get 3 of 4 questions to level up!

Solving by taking the square root

• Solving quadratics by taking square roots (Opens a modal)
• Solving quadratics by taking square roots examples (Opens a modal)
• Quadratics by taking square roots: strategy (Opens a modal)
• Solving quadratics by taking square roots: with steps (Opens a modal)
• Solving simple quadratics review (Opens a modal)
• Quadratics by taking square roots (intro) Get 3 of 4 questions to level up!
• Quadratics by taking square roots Get 3 of 4 questions to level up!
• Quadratics by taking square roots: strategy Get 3 of 4 questions to level up!
• Quadratics by taking square roots: with steps Get 3 of 4 questions to level up!

Vertex form

• Vertex form introduction (Opens a modal)
• Graphing quadratics: vertex form (Opens a modal)
• Quadratic word problems (vertex form) (Opens a modal)
• Graph quadratics in vertex form Get 3 of 4 questions to level up!
• Quadratic word problems (vertex form) Get 3 of 4 questions to level up!

Solving quadratics by factoring

• Solving quadratics by factoring (Opens a modal)
• Solving quadratics by factoring: leading coefficient ≠ 1 (Opens a modal)
• Solving quadratics using structure (Opens a modal)
• Quadratic equations word problem: triangle dimensions (Opens a modal)
• Quadratic equations word problem: box dimensions (Opens a modal)
• Solving quadratics by factoring review (Opens a modal)
• Quadratics by factoring (intro) Get 3 of 4 questions to level up!
• Quadratics by factoring Get 3 of 4 questions to level up!
• Solve equations using structure Get 3 of 4 questions to level up!

The quadratic formula

• The quadratic formula (Opens a modal)
• Understanding the quadratic formula (Opens a modal)
• Worked example: quadratic formula (example 2) (Opens a modal)
• Worked example: quadratic formula (negative coefficients) (Opens a modal)
• Using the quadratic formula: number of solutions (Opens a modal)
• Quadratic formula review (Opens a modal)
• Discriminant review (Opens a modal)
• Quadratic formula Get 3 of 4 questions to level up!
• Number of solutions of quadratic equations Get 3 of 4 questions to level up!

Completing the square intro

• Completing the square (Opens a modal)
• Worked example: Completing the square (intro) (Opens a modal)
• Worked example: Rewriting expressions by completing the square (Opens a modal)
• Worked example: Rewriting & solving equations by completing the square (Opens a modal)
• Completing the square (intro) Get 3 of 4 questions to level up!
• Completing the square (intermediate) Get 3 of 4 questions to level up!

More on completing the square

• Solve by completing the square: Integer solutions (Opens a modal)
• Solve by completing the square: Non-integer solutions (Opens a modal)
• Worked example: completing the square (leading coefficient ≠ 1) (Opens a modal)
• Solving quadratics by completing the square: no solution (Opens a modal)
• Proof of the quadratic formula (Opens a modal)
• Solving quadratics by completing the square (Opens a modal)
• Completing the square review (Opens a modal)
• Quadratic formula proof review (Opens a modal)
• Solve equations by completing the square Get 3 of 4 questions to level up!
• Completing the square Get 3 of 4 questions to level up!

Strategizing to solve quadratic equations

• Strategy in solving quadratic equations (Opens a modal)
• Strategy in solving quadratics Get 3 of 4 questions to level up!

Quadratic standard form

• Finding the vertex of a parabola in standard form (Opens a modal)
• Graphing quadratics: standard form (Opens a modal)
• Quadratic word problem: ball (Opens a modal)
• Graph quadratics in standard form Get 3 of 4 questions to level up!
• Quadratic word problems (standard form) Get 3 of 4 questions to level up!

Features & forms of quadratic functions

• Forms & features of quadratic functions (Opens a modal)
• Worked examples: Forms & features of quadratic functions (Opens a modal)
• Vertex & axis of symmetry of a parabola (Opens a modal)
• Finding features of quadratic functions (Opens a modal)
• Interpret quadratic models: Factored form (Opens a modal)
• Interpret quadratic models: Vertex form (Opens a modal)
• Graphing quadratics review (Opens a modal)
• Creativity break: How does creativity play a role in your everyday life? (Opens a modal)
• Features of quadratic functions: strategy Get 3 of 4 questions to level up!
• Features of quadratic functions Get 3 of 4 questions to level up!
• Graph parabolas in all forms Get 3 of 4 questions to level up!
• Interpret quadratic models Get 3 of 4 questions to level up!

Comparing quadratic functions

• Comparing features of quadratic functions (Opens a modal)
• Comparing maximum points of quadratic functions (Opens a modal)
• Compare quadratic functions Get 3 of 4 questions to level up!

Transforming quadratic functions

• Intro to parabola transformations (Opens a modal)
• Shifting parabolas (Opens a modal)
• Scaling & reflecting parabolas (Opens a modal)
• Quadratic functions & equations: FAQ (Opens a modal)
• Shift parabolas Get 3 of 4 questions to level up!
• Scale & reflect parabolas Get 3 of 4 questions to level up!

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Unit 3 – Functions

In this unit we review the basic concept of a function and emphasize multiple representations of these foundational tools.  Graphical features of functions, including maximums, minimums, intervals of increase and decrease along with domain and range are introduced.  Classic function notation is used throughout the unit.  Average rate of change is introduced as a tool for measuring the growth or decline in a function.  We hope that visitors will use these lessons and give us feedback to make them better.

Introduction to Functions

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

Function Notation

Graphs of Functions

Graphical Features and Terminology

Exploring Functions Using the Graphing Calculator

Average Rate of Change

The Domain and Range of a Function

Unit Review

Unit #3 Review – Functions

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit #3 Assessment Form A

EDITABLE ASSESSMENT

Unit #3 Assessment Form B

Unit #3 Assessment Form C

Unit #3 Assessment Form D

Unit #3 Exit Tickets

Unit #3 Mid-Unit Quiz (Through Lesson #3).Form A

Unit #3 Mid-Unit Quiz (Through Lesson #3).Form B

Unit #3 Mid-Unit Quiz (Through Lesson #3).Form C

U03.AO.01 – Lesson #6.5 – Motion and Average Rate of Change

EDITABLE RESOURCE

U03.AO.02 – More Work with Domain and Range

U03.AO.03 – Average Rate of Change Extra Practice

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Multi-Step Equations Exercises

Multi-step equations practice problems with answers.

For this exercise, I have prepared seven (7) multi-step equations for you to practice. If you feel the need to review the techniques involved in solving multi-step equations, take a short detour to review my other lesson about it. Click the link below to take you there!

Solving Multi-Step Equations

1) Solve the multi-step equation for [latex]\large{c}[/latex].

[latex]c – 20 = 4 – 3c[/latex]

Add both sides by [latex]20[/latex]. Next, add [latex]3c[/latex] to both sides. Finally, divide both sides by the coefficient of [latex]4c[/latex] which is [latex]4[/latex] to get [latex]c=6[/latex].

2) Solve the multi-step equation for [latex]\large{n}[/latex].

[latex] – \,4\left( { – 3n – 8} \right) = 10n + 20[/latex]

• Remember to always perform the same operation on both sides of the equation.
• Subtract by [latex]32[/latex].
• Subtract by [latex]10n[/latex].
• Divide by [latex]2[/latex]
• The final solution is [latex]n=-6[/latex].

3) Solve the multi-step equation for [latex]\large{y}[/latex].

[latex]2\left( {4 – y} \right) – 3\left( {y + 3} \right) = – 11[/latex]

Apply twice the Distributive Property of Multiplication over Addition to the left side of the equation. Then combine like terms . Add both sides by [latex]1[/latex] followed by dividing both sides of the equation by [latex]-5[/latex].

4) Solve the multi-step equation for [latex]\large{k}[/latex].

[latex]{\Large{{6k + 4} \over 2}} = 2k – 11[/latex]

Multiply both sides by [latex]2[/latex]. Next, subtract [latex]4[/latex] to both sides. Then, subtract [latex]4k[/latex]. Finally, divide by [latex]2[/latex] to obtain the value of [latex]k[/latex] which is [latex]-13[/latex].

5) Solve the multi-step equation for [latex]\large{x}[/latex].

[latex] – \left( { – 8 – 3x} \right) = – 2\left( {1 – x} \right) + 6x[/latex]

Apply the Distributive Property on both sides of the equation. Be careful when multiplying expressions with the same or different signs . Next, add [latex]2[/latex] to both sides, then subtract [latex]3x[/latex], and finally finish it off by dividing [latex]5[/latex] to both sides.

6) Solve the multi-step equation for [latex]\large{m}[/latex].

[latex]{\large{3 \over 4}}m – 2\left( {m – 1} \right) = {\large{1 \over 4}}m + 5[/latex]

7) Solve the multi-step equation for [latex]\large{x}[/latex].

[latex]3\left( {3x – 8} \right) – 5\left( {3x – 8} \right) = 4\left( {x – 2} \right) – 6\left( {x – 2} \right)[/latex]

You may also be interested in these related math lessons or tutorials:

Two-Step Equations Practice Problems with Answers

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