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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Quadratic Relations and Their Graphs: Unit Review

1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4About 40 minutes
Learning goals
  • I can recognize, graph and describe quadratic relations and their key features.
  • I can compare y = x² and y = 2ˣ and evaluate zero and negative exponents.
  • I can explain the roles of a, h and k and sketch y = a(x − h)² + k.
  • I can find the equation of a parabola in vertex form.

Words to know

parabola
The graph of a quadratic relation.
second differences
Differences of the first differences; constant and non-zero for a quadratic relation.
vertex
The turning point of a parabola.
axis of symmetry
The vertical line x = h through the vertex.
zeros
The x-intercepts of a graph.
vertex form
y = a(x − h)² + k, with vertex (h, k).
transformation
A translation, reflection, stretch or compression of a graph.

Let's review!

In this unit you collected quadratic data and drew curves of best fit, used first and second differences to recognize quadratic relations, graphed y = ax² + bx + c and named its key features (vertex, axis of symmetry, zeros, y-intercept, maximum or minimum value), used those features to solve problems, and compared y = x² with y = 2ˣ (with zero and negative exponents). Then you learned how a, h and k transform y = x², sketched y = a(x − h)² + k by hand, and found vertex-form equations from graphs, tables and real situations.

This review has a few questions from every lesson. Try each one on paper, then press Show answer to check.

How to use this review

Do one or two sections at a time rather than all at once. Write full solutions, not just answers, and draw a quick sketch whenever a question is about a graph. For every question you get wrong, write down why (sign error? mixed up h? forgot to multiply by a?) and go back to that lesson's worked examples. Keep Desmos handy to check graphs. Finish with the 10-question quiz.

y =a(x −h)² +ka: shape and directiona > 0: opens upa < 0: opens downsize > 1: stretchsize 0 to 1: compressiona < 0: reflection in x-axish: left or righth > 0: moves righth < 0: moves leftvertex x-coordinate haxis of symmetry x = hk: up or downk > 0: moves upk < 0: moves downvertex y-coordinate kmax or min value k
The big picture: in y = a(x − h)² + k, a controls shape and direction, h moves the parabola left or right, and k moves it up or down. The vertex is (h, k).

Collecting Data That Curves: Graphs and Curves of Best Fit

1

A ball rolls from rest: d = 0, 5, 20, 45 cm at t = 0, 1, 2, 3 s. Is a line or a curve the better model? Predict d at t = 4 s.

2

Give two features of a good curve of best fit.

3

Estimating between data points is ______; beyond the data is ______. Which is more reliable?

First and Second Differences: Recognizing a Quadratic Relation

1

Classify: x = 1, 2, 3, 4, 5 and y = 2, 5, 10, 17, 26.

2

What is the second difference of y = −4x² + x + 3 (x-steps of 1)?

3

Find the next value of the quadratic pattern 0, 3, 10, 21, …

Graphing y = ax² + bx + c: Every Quadratic Makes a Parabola

1

Does y = −0.2x² + 3x open up or down? What is its y-intercept?

2

Evaluate y = −x² + 3 at x = −2.

3

Is y = (x + 1)² − x² quadratic? Explain.

Key Features of a Parabola: Vertex, Axis of Symmetry, Zeros and Intercepts

1

Look at parabola P in the graph below. State its vertex, axis of symmetry, zeros, y-intercept and minimum value.

2

y = x² + 4x − 5 has zeros −5 and 1. Find the vertex.

3

A parabola has vertex (3, −2) and passes through (1, 6). Name another point on it.

xy−2−10123456−3−2−10123456parabola P
Parabola P, used in the review questions for Key Features and Finding the Equation. The marked points are (−1, 2.5), (0, 0), (2, −2), (4, 0) and (5, 2.5).

Using Parabola Features in Context: Maximum and Minimum Values

1

h = −5t² + 20t models a ball's height (m). When does it land and what is its maximum height?

2

A rectangle has a perimeter of 24 m. Find the largest possible area.

3

A diver's height model has zeros t = −0.6 and t = 2.1. Which one answers "When does the diver enter the water?"

y = x² Versus y = 2ˣ: Comparing Graphs, and Zero and Negative Exponents

1

Evaluate 2⁻⁵ and (3/4)⁻¹.

2

Evaluate (−4)⁰ and −4⁰.

3

Which is bigger at x = 6: x² or 2ˣ? Name one feature y = 2ˣ has that y = x² does not.

Moving Up and Down: The Graph of y = x² + k

1

Describe y = x² − 6 and state its vertex and number of zeros.

2

Where does (3, 9) on y = x² go on y = x² − 6?

3

Find k if y = x² + k passes through (2, 10).

Just for fun

Why did the parabola do so well on its unit test?

It knew its ups and downs, stayed symmetric under pressure, and always reached its maximum potential.

Moving Left and Right: The Graph of y = (x − h)²

1

State the vertex and y-intercept of y = (x + 5)².

2

Write the equation of y = x² translated 1.5 units right.

3

Are y = (x − 2)² and y = x² − 2 the same? Test x = 0.

Changing the Shape: Stretches, Compressions and Reflections of y = ax²

1

Describe y = −0.5x² and give the points at x = ±2.

2

A parabola with vertex (0, 0) passes through (3, 27). Find a.

3

Give the step pattern for y = 3x².

Vertex Form y = a(x − h)² + k: The Roles of a, h and k

1

For y = 4(x − 2)² − 7, state the vertex, axis, minimum value and y-intercept.

2

Describe the transformations in y = −(x + 1)² + 3 and say how many zeros it has.

3

Use the mapping rule to find the image of (1, 1) under y = 2(x + 3)² − 4.

Sketching y = a(x − h)² + k by Hand

1

List the vertex and four more points to sketch y = −(x − 1)² + 4.

2

What are the first two steps from the vertex for y = 0.5(x + 2)² − 1?

3

Sketch y = (x + 2)² − 4 and state its zeros.

Finding the Equation of a Parabola in Vertex Form

1

Find the equation of parabola P in the graph above.

2

A parabola has vertex (2, 1) and passes through (4, 9). Find its equation.

3

A parabola has zeros −1 and 5 and a maximum value of 18. Find its equation.

Check your understanding

Quiz

1.

Which table is quadratic? (x = 0, 1, 2, 3)

2.

The zeros of a parabola are −6 and 2. Its axis of symmetry is…

3.

h = −5t² + 30t models a ball's height in metres. Its maximum height is…

4.

What is 5⁻²?

5.

Which feature does y = 2ˣ have?

6.

y = (x + 4)² − 1 is y = x² translated…

7.

Which parabola is wider than y = x²?

8.

The vertex and direction of y = −2(x − 5)² + 3 are…

9.

From the vertex of y = 2(x − 1)² + 3, going 1 right takes you to…

10.

A parabola has vertex (−1, 4) and passes through (1, 0). Its equation is…

For parents and teachers

How to use the results: each section of the review matches one lesson. If your teen makes a particular kind of mistake, revisit that lesson:

  • Joining dots, mixing up interpolation and extrapolation → Collecting Data That Curves.
  • Errors in first/second differences or classifying tables → First and Second Differences.
  • Squaring the negative in −x², wrong direction of opening → Graphing y = ax² + bx + c.
  • Confusing vertex, axis (x = …) and max/min value; finding the axis from zeros → Key Features of a Parabola.
  • Keeping negative times, missing units in word problems → Using Parabola Features in Context.
  • Thinking 2⁻³ is negative or 2⁰ = 0; mixing up x² and 2ˣ → y = x² Versus y = 2ˣ.
  • Moving the wrong way for k or h → y = x² + k and y = (x − h)².
  • Calling a stretch "wider", sign of a → Stretches, Compressions and Reflections.
  • Reading (h, k) with the wrong sign, mapping rule errors → Vertex Form.
  • Steps from the origin instead of the vertex, forgetting to multiply by a → Sketching by Hand.
  • Substituting the vertex to find a, sign errors → Finding the Equation.

The companion unit Quadratic Expressions and Equations builds on this one: expanding and factoring, and solving quadratic equations to find zeros exactly.