DECK 2: QUADRATIC TERMS INTRO Flashcards
What is the shape of a quadratic graph called?
A parabola
What is the highest or lowest point (x,y) called on a parabola? It is on the AOS. It is sometimes written as (h,k)
The vertex.
Is the vertex of a parabola always on the axis of symmetry?
YES. The x value of the vertex is the axis of symmetry.
What is the axis of symmetry?
It is the vertical line that goes through the middle of the parabola- parabola folds on itself. THE VERTEX IS ON IT
What does VERTICAL mean?
Up and Down
What does the line x = 10 look like?
A vertical line through 10 on the x axis
What is the equation of a vertical line through 7 on the x axis? (visualize a vertical line through 7 on the x axis)
x = 7
All vertical lines have equation
“ X = some number”
Since the axis of symmetry is a vertical line, the equation of the axis of symmetry is always:
X = some number
you have to write “x = ? “
What does HORIZONTAL mean?
Left and right
Do all parabolas have x-intercepts? If so, how many?
NO - they may have
NONE, ONE or TWO.
Do all parabolas have y-intercepts? If so, how many?
YES, they all have EXACTLY ONE y intercept.
How can a parabola have no real roots (no x intercepts)?
If it doesn’t touch the x axis. Smiling parabola above, or frowning parabola below.
How can a parabola have only one real root (one x intercept)
If it touches the x-axis only at one point (the vertex) and “bounces off”
How can a parabola have two real roots (two x intercepts)
If it crosses twice.
How do you find the y intercept for any linear or quadratic function?
Set X=0 and solve for y
How do you find the x-intercepts of any linear or quadratic functions?
Set Y=0 and solve for x (this involves setting each factor = 0 and solving for x for each)
What are the three forms of quadratic functions?
standard, vertex and factored (intercept) forms
What form?
f(x)=ax 2 +bx+c
Standard
What is the standard form of a quadratic?
f(x)=ax 2 +bx+c
What form?
f(x) = a(x-h) 2 +k
Vertex form
What is the vertex form of a quadratic?
f(x) = a(x-h) 2 +k
What form?
f(x)=a(x-Z)(x-Z)
Intercept or Factored form
What is the intercept (factored) form of a quadratic?
f(x)=a(x-Z)(x-Z) where Z and Z are ZEROS, roots, x-intercepts
For any polynomial in factored form, how do you find the ROOTS, ZEROS, SOLUTIONS, X-INTERCEPTS?
SET THE FACTORS EQUAL TO ZERO AND SOLVE!
EX: f(x)=3 (x-5)(x+10) Roots are 5 and -10
because (x-5)=0 when x=5
and (x+10)=0 wnen x=-10
You can often do the math in your head. But if a factor is (3x-5), you would have to solve for
x in 3x-5=0
What does “a” tell us about the graph?
Two things. If it is + it smiles, if it is - it frowns AND if a>1 (bigger than 1) it gets narrow. If a <1 (smaller than 1) it is WIDE
Is “a” the same for all three forms?
Yes, whatever “a” is in standard form, it will be the same value in factored and in vertex form. It always tells us + SMILE, -FROWN,
bigger than 1 it gets NARROW, and smaller than one it gets WIDE
What if you have +a value?
Concave UP, smiling
What if you have -a value?
Concave DOWN, frowning
How do you know when a parabola is smiling?
when a is positive
How do you know when a parabola is frowning?
when a is negative
What are the three forms of a quadratic function and their equations?
Standard y=ax 2 +bx+c Vertex y=a(x-h) 2 +k Factored y=a(x+f)(x+g)
What is the GIFT you get in standard form?
The y intercept. It is at C or (0,c)
What is the GIFT that you get in vertex form?
The vertex (h,k). This also give you the AOS and the MAX/MIN
What is the GIFT that you get in factored form?
The x intercepts - the ZEROS- the ROOTS.
WHAT FORM and WHAT IS THE GIFT?
f(x)= -2x2-4x+9
Standard Form
GIFT
Y intercept at (0,9)
WHAT FORM and WHAT IS THE GIFT?
f(x)= -2(x+7)2+11
Vertex Form
GIFT
Vertex at ( -7, 11)
WHAT FORM and WHAT IS THE GIFT?
f(x)= -2(x+9)(x-11)
Intercept/Factored Form
GIFT
X-intercepts at -9 and 11
or (-9,0) and (11,0)
f(x)=2x 2 -4x+6
What is the y intercept?
(0,6)
f(x)=2(x-5)(x-3)
What are the x intercepts?
At 5 and at 3
f(x)= 2(x+4) 2 -3
What is vertex?
(-4, -3)
If you have a vertex, like (5.10), what else do you have?
The AOS and MAX/MIN.
The AOS is at x=5 and the max/min is y=10
y=-2x 2 -4x+6
concave up or concave down?
a= -2 concave DOWN
f(x)=2(x-5)(x-3)
concave up or concave down?
a=2 concave UP
f(x)= -3(x+4) 2 -3
concave up or concave down?
a= -3 concave DOWN
If we know the coordinates of the vertex is always (AOS, MAX/MIN), then if a vertex is at (5,12), WHAT OTHER INFORMATION DO WE KNOW?
We know the AOS is at x=5
and the max/min is at y=12
f(x) = a(x-h) 2 + k
How do we know if it opens up or down?
by the sign of a, if its positive, up, negative , down
If a parabola is concave up, smiling, will it have a MAX or MIN?
IF it is smiling, it goes up forever, so it has no max value IT ONLY HAS A MIN Y VALUE
If a parabola is concave down, frowning, will it have a MAX or MIN?
If frowning, it goes down forever so it has no minimum, IT ONLY HAS A MAX Y VALUE
f(x)= 2(x+4) 2 -3
What is vertex ?
at (-4, -3), SMILING so it opens upwards, -3 is a MIN value
f(x)= (x+9)2-8
What is vertex
at (-9, -8)
f(x)=2(x-5)(x-3)
What are the ROOTS?
5 and 3
y=2(x-7)(x+3)
what are the coordinates of the x intercepts?
(7,0) and (-3,0)
How do we know when to take opposites when working with quadratics?
usually when they are in parentheses
Suppose a vertex of a parabola is at (5,10), what else do you know?
You know the AOS is at x=5
and the max/min is at y=10.
If a vertex is at (8, -9) and frowning, what do you know?
You know the AOS is at x=8
and the MAX value is y=-9
FROWNS HAVE MAXIMUM HEIGHTS BUT GO DOWN FOREVER
If a vertex is at (5, 12) and smiling, what do you know?
You know the AOS is at x=5
and the MIN value is y=12
SMILES HAVE MINIMUM HEIGHTS BUT GO UP FOREVER
What is the word for a line that goes left and right?
Horizontal line
What is the word for a line that goes up and down?
Vertical line
What is the equation of a horizontal line through the y axis that is up 5
y=5
What is the equation of the vertical line that is over to the left at -10 on the x axis?
x= -10
what does “a” tell us in this quadratic?
f(x)= -5(x-3)2+13
TWO THINGS
It is - so frowning
5 makes it NARROW
What does “a” tell us in this quadratic?
f(x)= 0.25(x-3)2+13
TWO THINGS
It is + so smiling,
and 0.25 makes it wide.
