212. Graphing Monic Quadratic
Learning Intentions
- Generate tables of values for monic quadratic functions.
- Plot points accurately on the Cartesian plane.
- Draw parabolas Use symmetry and intercepts.
Pre-requisite Summary
- Understand substitution into algebraic expressions.
- Recall order of operations when evaluating expressions.
- Recognise coordinates are written as
. - Plot ordered pairs on a Cartesian plane.
- Identify the x-axis and y-axis.
- Understand that a monic quadratic has leading coefficient
. - Recall that quadratic graphs are called parabolas.
- Understand symmetry in graphs.
Worked Examples
Worked Example 1
Generate a table of values for the quadratic function:
using the values:
Worked Example 2
Generate a table of values for:
for:
Then identify the x-intercepts.
Worked Example 3
Plot the points from the function:
using:
Then sketch the parabola.
Worked Example 4
Sketch the graph of:
by:
a) Finding the x-intercepts
b) Finding the y-intercept
c) Using symmetry to complete the sketch
Worked Example 5
For the quadratic:
a) Generate a table of values
b) Plot the points
c) Sketch the parabola
Problems
Problem 1
Generate a table of values for:
using:
Problem 2
Generate a table of values for:
for:
Then identify the x-intercepts.
Problem 3
Plot the points from:
using:
Then sketch the parabola.
Problem 4
Sketch the graph of:
by:
a) Finding the x-intercepts
b) Finding the y-intercept
c) Using symmetry to complete the sketch
Problem 5
For the quadratic:
a) Generate a table of values
b) Plot the points
c) Sketch the parabola
Exercises
Understanding and Fluency
Exercise 1
Generate a table of values for:
for:
Exercise 2
Generate a table of values for:
for:
Exercise 3
Generate a table of values for:
for:
Exercise 4
Plot the points for:
using:
Exercise 5
Plot the points for:
using:
Exercise 6
Sketch the parabola:
Exercise 7
Sketch the parabola:
Exercise 8
Identify the x-intercepts and y-intercept of:
Exercise 9
Identify the axis of symmetry for:
Exercise 10
Generate a table of values and sketch:
Reasoning
Exercise 11
Explain why the graph of:
is symmetrical about the y-axis.
Exercise 12
A student claims the graph of:
has no x-intercepts.
Explain whether the student is correct.
Exercise 13
Describe how the graph changes when comparing:
and
Exercise 14
Explain how symmetry can help reduce the number of calculations needed when sketching a parabola.
Problem-solving
Exercise 15
A basketball is modelled by:
a) Find the intercepts
b) Sketch the parabola
c) Determine whether the graph opens upward or downward
Exercise 16
The height of a water fountain is modelled by:
a) Generate a table of values
b) Sketch the parabola
c) Estimate the minimum point from the graph
Exercise 17
A designer creates an arch shaped by:
a) Find where the graph crosses the x-axis
b) Sketch the graph
c) Describe the symmetry of the parabola
Potential Misunderstandings
- Thinking all quadratic graphs are symmetrical about the y-axis.
- Forgetting to square negative numbers correctly.
- Mixing up coordinates as
instead of . - Plotting points between grid intersections inaccurately.
- Assuming all parabolas cross the x-axis.
- Forgetting that monic quadratics open upward when the coefficient of
is positive. - Confusing the y-intercept with the turning point.
- Drawing parabolas with sharp corners instead of smooth curves.
- Forgetting to use symmetry to check plotted points.