IB Math Studies Yr 1
Name_________________________________
Date_______________
5-1 Properties of a Quadratic Function
Today’s Learning Goals:
#1 How do we determine the axis of symmetry and vertex of a quadratic function?
Properties of Quadratic functions

A quadratic equation is an equation in the form of y= ax2 + bx + c, where the highest degree is 2.

The graph of a quadratic equation is called a parabola.

In mathematics, a parabola is a plane curve which is symmetrical and is approximately U-shaped.

The parabolas either open __________or __________.

The turning point of the parabola is called the ________________.

The vertex can either be a _______________ or _________________ value.

The vertical line that cuts the parabola into two equal halves is called the __________________________.

The axis of symmetry always passes through the ________________ of the parabola.
DIRECTIONS: Based on the definitions above, label the vertex, axis of symmetry, x-intercepts, and y-intercept
of the parabola below.
IB Math Studies Yr 1
FINDING THE PROPERTIES OF A PARABOLA WHEN GIVEN THE GRAPH

For each of the parabolas graphed below, please find the following:
a. The equation of the axis of symmetry
b. The coordinates of the vertex
c. The coordinates of the y-intercept
IB Math Studies Yr 1
FINDING THE Y-INTERCEPT WHEN GIVEN THE EQUATION

What is the y-intercept of a graph?

How can you find the coordinates of the y-intercept when given the equation of a parabola?
1. Consider the quadratic function 𝑓(𝑥) = 3 + 4𝑥 − 𝑥 2 .
(a) Find the coordinates of the y-intercept of the graph of 𝑓.
2. The graph of the quadratic function 𝑓(𝑥) = −(𝑥 − 5)(𝑥 + 1) intersects the y-axis at point A.
(a) Find the coordinates of point A.
IB Math Studies Yr 1
FINDING THE AXIS OF SYMMETRY AND VERTEX WHEN GIVEN THE EQUATION

The axis of symmetry is the vertical line that cuts the parabola into two equal halves. To find the axis of
symmetry, you can use the axis of symmetry formula that is given to you in the IB Formula Booklet.

Before you use the axis of symmetry formula, first make sure the quadratic equation is expressed in
standard form.
The axis of symmetry always passes through the vertex of a parabola. This means that the axis of
symmetry is always the x-value of the coordinates of the vertex.

3. Consider the quadratic function 𝑓(𝑥) = −𝑥 2 + 4𝑥 + 5.
(a) Find the equation of the axis of symmetry.
(b) State the coordinates of the vertex.
IB Math Studies Yr 1
4. Consider the quadratic function 𝑓(𝑥) = 3 + 4𝑥 − 𝑥 2 .
(a) Find the equation of the axis of symmetry.
(b) State the coordinates of the vertex.
5. The graph of the quadratic function 𝑓(𝑥) = 22 + 7𝑥 − 2𝑥 2 has its vertex at point B .
(a) Find the coordinates of B.
IB Math Studies Yr 1
PUTTING IT ALL TOGETHER
6. The graph of the quadratic function 𝑓(𝑥) = 5 − 12𝑥 − 2𝑥 2 intersects the y-axis at point A and has its
vertex at point B .
(a) Find the equation of the axis of symmetry.
(b) Find the coordinates of B.
(c) Find the coordinates of A.
IB Math Studies Yr 1
Name______________________________________
Date_______________
5-1 Homework
1. For each of the parabolas graphed below, please find the following:
 The equation of the axis of symmetry
 The coordinates of the vertex
 The coordinates of the y-intercept
a)
b)
c)
IB Math Studies Yr 1
2. Consider the quadratic function 𝑓(𝑥) = −6 + 𝑥 − 𝑥 2 .
(a) Find the coordinates of the y-intercept of the graph of 𝑓.
3. Consider the quadratic function 𝑓(𝑥) = −10𝑥 + 2𝑥 2 .
(a) Find the equation of the axis of symmetry.
(b) State the coordinates of the vertex.
4. The graph of the quadratic function 𝑓(𝑥) = 8 + 16𝑥 − 4𝑥 2 intersects the y-axis at point A and has its
vertex at point B .
(a) Find the equation of the axis of symmetry.
(b) Find the coordinates of B.
(c) Find the coordinates of A.
IB Math Studies Yr 1
Name_________________________________
Date_______________
5-2 Finding the Intercepts of a Quadratic
Today’s Learning Goals:
#2 How do I graph a quadratic function? How many points do I need to graph a quadratic function?
#3 How do we find the x and y-intercepts of a quadratic function from the calculator?
#4 How do we find the vertex of a quadratic function from the calculator?
#5 What are the roots of a quadratic function, and how do I find them?
Do Now:
1. The following diagram shows part of the graph of y = -8x + x2 – 20.
a. Find the equation of the axis of symmetry.
b. Find the coordinates of the vertex.
c. Find the coordinates of the y-intercept.
Consider the function: 𝑦 = 𝑥2 – 2𝑥 – 3
Using the graph, answer the following questions...
1) What is the equation of this axis of symmetry?
2) Identify the coordinates of the vertex. Is this a maximum or
minimum?
3) Determine the coordinates of the y intercepts
We also call the x-intercepts:
4) Determine the coordinates of both the intercepts.
IB Math Studies Yr 1
NOTES ON ROOTS:
➢ Roots are also sometimes referred to as the solutions of the quadratic function
➢ If asked for the roots or solutions of a quadratic you can state x = _______ and x = ________
➢ If asked for the coordinates – your answer must be a coordinate.
To determine the
x-intercept graphically…
➢ If we are given a graph, we look to find the point where ________________________________
➢ If we are not given a graph, we must graph in in our calculators:
1.
and then type in your quadratic equation.
2. Graph the equation by hitting
3. Hit
6: ZStandard
2: Zero
4. Left bound?: Scroll with left and right arrow keys to the left of the intercept
5. Right bound? Scroll to the right of the intercept
6. Guess? Hit enter
7. It will give you the coordinate of that x-intercept
8. Repeat step 4 for the other x-intercept.
You try!
1) Find the coordinates of the x-intercepts of: 𝑦 = −𝑥 2 − 6𝑥 − 3
(Equation shown: 𝑦 = 𝑥 2 + 2𝑥 − 6)
Let’s Try Some:
1. Find the roots of the quadratic function: 𝑓(𝑥) = 7𝑥 + 𝑥 2 + 12
2. Find the solutions of the following: 𝑓(𝑥) = 𝑥 2 − 2𝑥 − 15
3. Find the coordinates of the x – intercepts of the function 𝑦 = 3𝑥 2 + 4𝑥 − 8
IB Math Studies Yr 1
Directions: Use the notes page to complete the following! For each quadratic equation you will find the
Equation of the Axis of Symmetry, the y-intercept, and x-intercepts or roots
1. 𝑦 = −𝑥 2 + 4𝑥 − 3
Equation of the Axis of Symmetry
y-intercept
roots
y-intercept
Coordinates of x-intercepts
y-intercept
solutions
2. 𝑦 = −𝑥 2 + 2𝑥 + 1
Equation of the Axis of Symmetry
3. 𝑦 = 18𝑥 + 3𝑥 2 + 9
Equation of the Axis of Symmetry
IB Math Studies Yr 1
4. 𝑦 = 3 + 3𝑥 2
Equation of the Axis of Symmetry
y-intercept
Coordinates of x-intercepts
y-intercept
x-intercepts
y-intercept
roots
5. 𝑦 = 5 + 𝑥 2 − 6x
Equation of the Axis of Symmetry
6. 𝑦 = −8𝑥 + 12 + 𝑥 2
Equation of the Axis of Symmetry
IB Math Studies Yr 1
Name_____________________________
Date _____________________________
Lesson 5-2: Homework
2. Sketch the graph of the parabola 𝑦 = −2𝑥 2 + 4𝑥 + 6 on the set of axes below.
(a) Write down the coordinates of the local maximum or minimum.
(b) Write down the coordinates of the y-intercept.
(c) Write down the coordinates of the x-intercepts.
(d) Write down the equation of the axis of symmetry.
3. Sketch the graph of the parabola 𝑦 = 𝑥 2 + 𝑥 − 12 on the set of axes below
(a) Write down the equation of the axis of symmetry.
(b) Write down the coordinates of the local maximum or minimum.
IB Math Studies Yr 1
Directions: For #3-4, find the equation of the axis of symmetry, the coordinates of the vertrex, and the
coordinates of the y-intercept.
4. 𝑦 = −3𝑥 2 + 6𝑥 − 9
Equation of the Axis of Symmetry
Coordinates of Vertex
Coordinates of Y-Intercept
Coordinates of Vertex
Coordinates of Y-Intercept
5. 𝑦 = 𝑥 2 + 11𝑥
Equation of the Axis of Symmetry
6. The diagram below shows part of the graph of y = −x2 + 6x −7 . The line x = 3 is the axis of symmetry.
M and N are points on the curve, as shown.
y
Find the coordinates of
(a) M;
N
(b) N.
x
0
M
x=2
IB Math Studies Yr 1
Putting it All Together
7. The diagram shows the parabola 𝑦 = −𝑥 2 + 6𝑥 + 7. The points A and C are the x-intercepts and the point B is
the maximum point.
y
B
A
0
a. Write down the coordinates of the y-intercept.
b. Find the equation of the axis of symmetry of the graph above.
c. Find the coordinates of the maximum point, B, on the graph above.
d. Calculate the coordinates of the points A and C.
C
x
IB Math Studies Yr 1
Name_________________________________
Date_______________
5-3 Solving Systems of Equations with Quadratics
Today’s Learning Goals:
How can I solve a system of equations using the calculator ?
Warm Up: On the grid below, graph the lines 𝑥 = −4 and 𝑦 = 2.
a) How many solutions are there in this system of
equations?
b) What is the solution to this system of equations?
c) What are we looking for when we are asked to find
the solution of a system of equations?
Linear – Quadratic System


This familiar linear- quadratic system, where only one variable is squared in the quadratic, will be the
graph of a parabola and a straight line.
When a parabola and a straight line are graphed on the same set of axes, three situations are possible.
Number of Solutions:
Number of Solutions:
Number of Solutions:
Solution set:
Solution set:
Solution set:
IB Math Studies Yr 1
1. The graph of the function𝑦 = 𝑓(𝑥) is graphed to the right.
(a) On the same grid, graph 𝑦 = −3
(b) Write down the coordinates of the points of intersection.
(c) On the same grid, graph 𝑦 = −2
*How would you find the intersections between 𝒇(𝒙)
And 𝒚 = −𝟐?*
Finding Points of Intersection In the calculator!
1.
2.
3.
4.
5.
Plug our equations into our calculator for Y1 and Y2
Hit zoom → 6
Go to 2nd Trace
Select option 5: Intersect
Go to the one of the points of intersection using your
left and right arrow keys
6. Hit Enter 3 times!
7. Repeat steps 3, 4, 5, 6 if there is more than one intersection!
Example: Solve the following system of equations: 𝑓(𝑥) = 5 + 3𝑥 − 𝑥 2 and 𝑦 = 2𝑥 + 3
IB Math Studies Yr 1
1. The graph of the function 𝑓(𝑥) = −2𝑥 2 − 4𝑥 + 6 is shown below.
(a) On the same grid, graph 𝑦 = 5
(b) Write down the coordinates of the points of intersection.
2
2. The graph of the function 𝑓(𝑥) = 3 𝑥 2 − 4𝑥 + 7 is shown below.
Line 𝐿 intersects the y-axis at 6 and has a gradient of −2.
(a) On the same grid, graph line 𝐿.
(b) Write down the coordinates of the points of intersection.
IB Math Studies Yr 1
PUTTING IT ALL TOGETHER
 This is a question that combines content from Unit 4 and Unit 5
The graph of a line 𝐿 passing through (1,1) and (2, 3) and the graph of a parabola P
has the function 𝑃(𝑥) = 𝑥 2 − 3𝑥 − 4.
a) Find the gradient of line L.
b) Write the equation of line L in the form 𝑦 = 𝑚𝑥 + 𝑐.
The graphs of line L and parabola P intersect at approximately (5.54, 10.1).
c) State the coordinates of the other point of intersection.
IB Math Studies Yr 1
Name_________________________________
Date_______________
5-4 Determining C of Quadratic Equation
Today’s Learning Goals:
#1 How do we find the value of C for 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 by looking at the graph?
Warm-Up:
Answer the following questions in order to prepare for today’s lesson.
The graph of li ne 𝐿 is given below.
The equation for line 𝐿 can be represented by 𝑦 = 𝑚𝑥 + 𝑐.
1. Find the value of
(a)
c;
•
How can you find the value of “c”
by looking at the graph?
(b)
m.
•
How can you find the value of “m”
without using the gradient formula?
IB Math Studies Yr 1
• A standard form of a quadratic equation is _______________________.
•
In standard form, the value of “c” represents the ______________________ of the parabola.
➢ To find the value of “c”, look for where the graph crosses the _____________ or
a coordinate where __________.
Model Problem #1:
The graph of the quadratic function𝑓(𝑥) = 𝑐 + 4𝑥 − 𝑥 2 intersects the y-axis at point A(0, 5).
a. Write down the value of c.
b. Find the equations of the axis of symmetry
c. Find the coordinates of the vertex
d. Find the x intercepts.
IB Math Studies Yr 1
Model Problem #2:
The figure below shows part of the graph of a quadratic function 𝑦 = −2𝑥 2 + 4𝑥 + 𝑐.
y
8
a. Write down the value of c.
6
b. Find the coordinates of the vertex
4
2
–2
–1
1
2
3
c. Write down the roots of the function.
Model Problem #3:
The graph of a quadratic function has y-intercept -7 and one of its x-intercepts is -7. The x-coordinate of the
−13
vertex of the graph is 7 . The equation of the quadratic function is in the form 𝑦 = 2𝑥 2 + 13𝑥 + 𝑐.
a. Write down the value of c
b. Find the equation of the axis of symmetry
c.
Write down the second x-intercept of the function.
4 x
IB Math Studies Yr 1
1. The graph of the quadratic function 𝑓(𝑥) = −𝑥 2 + 4𝑥 + 𝑐 intersects the y-axis at point A(0,5).
a. Find the value of c
b. Find the coordinates of the maximum that
occurs at point B.
2. The following graph below shows equation 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 𝑐.
a. Find the value of c
b. Find the coordinates of the vertex
c. Write down the roots of the function
IB Math Studies Yr 1
3. The graph of a quadratic function has y-intercept -16 and one of its x-intercepts is 8. The x-coordinate of the
vertex of the graph is 3. The equation of the quadratic function is in the form 𝑦 = 𝑥 2 − 6𝑥 + 𝑐.
a. Write down the value of c
b. Write down the equation of the axis of symmetry
c. Write down the coordinates of the vertex.
d. Find the second x-intercept of the function.
4. The diagram shows the parabola y = – x2 + 6x + 7.The points A and C are the x-intercepts and the
y
point B is the maximum point.
a. Find the coordinates of A, B and C.
B
A
0
C
x
IB Math Studies Yr 1
Name_________________________________
Date_______________
5-5 Review of 5-1 through 5-4
REVIEW OF QUADRATIC FUNCTION PROPERTIES

What is the standard form of a quadratic
equation?

What is the axis of symmetry of a parabola?

When in standard form, what does “c”
represent?

What is the equation used to find the axis of
symmetry?

What is the vertex of a parabola?

How can you find the coordinates of the
vertex?

How can you find the points of intersection for
a linear-quadratic system of equations?

What are the roots of a parabola?

How can you find the coordinates of the
x-intercepts?
IB Math Studies Yr 1
IB QUESTIONS
1. The graph of the function 𝑦 = 𝑓(𝑥) is given to the right.
(a) Write down the coordinates of the y-intercept.
(b) Write the equation of the axis of symmetry for 𝑓(𝑥).
(c) Write down the coordinates of the local minimum for 𝑓(𝑥).
(d) State the coordinates of the x-intercepts.
(e) Graph the line 𝑦 = 5 on the same grid as 𝑓(𝑥).
(f) Write the coordinates of the points of intersection.
2. Consider the function 𝑦 = 2𝑥 2 − 6𝑥 + 5.
(a) Write down the coordinates of the y-intercept.
(b) Find the equation of the axis of symmetry of the function.
(c) Write down the coordinates of the local maximum or minimum of the function.
IB Math Studies Yr 1
3. The graph of a quadratic function has a y-intercept 10 and one of its x-intercepts is 1.
The x-coordinate of the vertex of the graph is −2.
The equation of the quadratic function is in the form 𝑦 = 𝑐 − 8𝑥 − 2𝑥 2 .
(a) Write down the value of c.
(b) Write the equation of the axis of symmetry.
(c) State the coordinates of the vertex.
(d) Find the coordinates of the second x-intercept.
4. The graph of the quadratic function 𝑓(𝑥) = 𝑐 + 2𝑥 − 0.5𝑥 2 . The point (0, 5) lies on the graph of 𝑓(𝑥).
(a) Write down the value of c.
(b) Find the coordinates of A.
Point B is the vertex of 𝑓(𝑥).
(c) Find the coordinates of B.
IB Math Studies Yr 1
5. The function f (x) is defined as f (x) = –x2 + 6x – 7. The diagram below shows part of the graph of f (x).
The maximum point on the curve is P (3, 2).
y
4
P(3, 2)
2
–1
1
2
3
4
5
6
x
–2
–4
–6
–8
–10
–12
(a) State the coordinates where the function f (x) crosses the x-axis.
The point Q lies on the curve and has coordinates (4, 1). A straight line L, passes through Q,
and has a and has a y-intercept of −1.
(a) Calculate the gradient of L.
(b) Find the equation of L.
(c) Graph the equation of L using the diagram above.
(d) The line L intersects the graph of f (x) twice. Find the x-coordinates where the two graphs intersect.