Vertex Form
Identify the transformations being applied to the parent quadratic function.
y=-3(x-4)^2+2
Reflected Across the X-axis
Vertically stretched by 3
Shifted right 4 and up 2
What is the vertex of the parabola?
y=x^2-4x+1
Vertex: (2, -3)
What is the parent quadratic function and what is its shape called?
y=x^2
Parabola
Factoring the following quadratic expression.
x^2+13x+42
(x+6)(x+7)
What are the two solutions to the equation
x(2x-3)=0
x=0
and
x=3/2
A man walks a certain distance due North and then the same distance plus a further 7 km due East. If the final distance from the starting point is 17 km, find the distances he walks North and East.
8 km north, 15 km east
What is the vertex of the following quadratic function?
y=(x+2)^2-7
Vertex: (-2, -7)
What is the y-intercept of the quadratic function?
y=2x^2+4x-4
Y-Intercept: (0, 4)
What is the following form of the quadratic function called?
y=a(x-h)^2+k
Vertex Form
Factoring the following quadratic expression.
3x^2+8x-3
(3x-1)(x+3)
What are the two zeros that come from the following binomials?
(x-3)(2x+5)=0
x=3, -5/2
A soccer ball is kicked. The ball's flight has a maximum height of 12.5m before landing 10m from where it started.
Write the equation to show the relation between height and distance and use this equation to determine how high the ball is after it has traveled a horizontal distance of 3m.
y=-1/2(x-5)^2+12.5
Height at 3m is 10.5m
What is the Axis of Symmetry for the equation
f(x) = 2x^2-12x+13
AoS:
x=3
Does the following parabola have a maximum or minimum value? Find that maximum or minimum!
y=-2x^2-8x+3
Vertex: (-2, 11)
The maximum value is 11
What is the Standard Form of the quadratic function?
y=ax^2+bx+c
Factoring the following expression.
x^3-2x^2-24x
x(x-6)(x+4)
Solve the quadratic equation.
x^2-2x-24=0
x=-4, 6
Explain what must be true for a parabola to have no x-intercepts?
The parabola doesn't cross the x-axis.
Either it's facing upwards and it's vertex is above the x-axis or it's facing downwards and it's vertex is below the x-axis.
Does the following quadratic function have a maximum or minimum value? What is that max or min y-value of the parabola?
y=-3(x+2)^2-4
Maximum Value of y=-4
What is the vertex of the parabola?
y=3x^2+18x+32
Vertex: (-3, 5)
Find the discriminant of 6x^2-2x-3 ?
What does this tell us about the number of x-intercepts?
Discriminant is 76.
This means it has 2 x-intercepts.
Rewrite the following quadratic equation in its Factored Form
y= -x^2+16x-55
y=-(x-5)(x-11)
Solve the quadratic equation:
-11x+3x^2+38 = -30+10x+2x^2
x=17 or x=4
A right triangle has two shorter sides that differ in length by 2 cm. The length of the hypotenuse is 4 cm more than the shortest side. Find the lengths of the three sides.
6 cm, 8 cm, 10 cm.
Does the following quadratic function have a maximum or minimum? What is that max or min value?
y=1/2x^2+3
Minimum Value of y=3
Find the two coordinates on the parabola below with y-values of 19.
y=1/4x^2+2x+19
The points are (-8,19) & (0,19)
Quadratics of the form 16x^2-49 are called what?
Factor 16x^2-49.
Difference of squares.
16x^2-49=(4x+7)(4x-7)
Factoring the following quadratic expression.
16x^2-40x+25
(4x-5)^2
Solve x^2+x-4=0 .
x=1.56
and
x=-2.56
A large fountain located in a park has 35 water jets. One of the streams of water follows a parabolic path so that its height is given by the equation h=-2x^2+6x+5 where h metres is the height of the stream of water at any distance x metres from its jet. What is the maximum distance the water jet can reach, rounded to the nearest tenth of a metre?
The maximum reach of the water jet is approximately 3.7 m.
Write the equation of a parabola with the given vertex.
Vertex: (-3, 7) that also passes through point (0, -11)
y=-2(x+3)^2+7
Write the quadratic relation in vertex form.
y=-2x^2-4x+6
y=-2(x+1)^2+8
Quadratics of the form 36x^2-60x+25 are called what?
Factor 36x^2-60x+25
Perfect square trimomials.
36x^2-60x+25=(6x-5)^2
Write the quadratic equation in standard form that has x-intercepts at (2,0) and (5,0) and also passes through the point (0,-3)
y=-3/10(x-2)(x-5)
y= -3/10x^2+21/10x-3
Between what two x values is the following quadratic positive (i.e. above the x-axis):
y=-x^2+7x-9
Between 1.7 and 5.3
A rocket is fired into the air from the edge of a cliff and falls to the ground. The height, h, in metres, of the rocket relative to the ground after t seconds is modelled by the equation h=-5t^2+10t+35 .
For how long is the rocket in the air before it hits the ground?
The rocket is in the air for approximately 3.8 seconds.