1.) Identify the axis of symmetry.

f(x) = x2 + 10x + 9

2.) Find the vertex for the parabola given below. f(x) = 3x2 + 12x + 16

3.) Find the quadratic function that includes each set of values.
(1, 0), (2, 5), (4, 21)

4.) f(x) = -2x2 - 3x + 35
Is this parabola concave up or concave down?

5.) A projectile is fired straight upward from the ground with an initial speed of 96 feet per second,then its height h after t seconds is given by the equation h(t) = -16t2 + 96t. Find the maximum height of the projectile.

#1

x^2+10x+9
= x^2+10x+25 + 8-25
= (x+5)^2 - 16
Since the vertex is at (-5,-16), the axis of symmetry is at x = -5

Or, you can consider that the roots are at -9 and -1, and the axis is midway between the roots, or at x = -5.

#2
3x^2+12x+16
= 3(x^2+4x) + 16
= 3(x^2+4x+4) + 16-3*4
= 3(x+2)^2 + 4
So, the vertex is at (-2,4)

#3
Plugging in values, we see that
a+b+c = 0
4a+2b+c = 5
16a+4b+c = 21
solve for a,b,c,and we see we have
x^2+2x-3

Or, since one root is at x=1,
y = a(x-1)(x-d)
5 = a(2-d)
21 = 3a(4-d)
a,d = 1,-3
y = (x-1)(x+3) = x^2+2x-3

#4 since the coefficient of x^2 is negative, the parabola opens downward

#5 since the vertex is at x = -b/2a, that will be when x = 3. h(3) = 144

1.) To find the axis of symmetry for a quadratic function, you can use the formula x = -b/2a.

For the given function f(x) = x2 + 10x + 9, the coefficient of x^2 is a = 1, and the coefficient of x is b = 10. Plugging these values into the formula, we get:

x = -(10) / (2 * 1)
x = -10 / 2
x = -5

Therefore, the axis of symmetry for the function f(x) = x2 + 10x + 9 is x = -5.

2.) The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.

To find the vertex for the given function f(x) = 3x2 + 12x + 16, we can use the formula:

h = -b/(2a)
k = f(h)

For this function, a = 3, b = 12, and c = 16. Plugging these values into the formulas, we get:

h = -(12) / (2 * 3)
h = -12 / 6
h = -2

Substituting h back into the equation to find k:

k = 3(-2)^2 + 12(-2) + 16
k = 12 - 24 + 16
k = 4

Therefore, the vertex of the function f(x) = 3x2 + 12x + 16 is (-2, 4).

3.) To find the quadratic function that includes a given set of values, you can use the points (x, y) to set up a system of equations.

Let's consider the points (1, 0), (2, 5), and (4, 21).

Using the general form of a quadratic function f(x) = ax^2 + bx + c, we can substitute the given points into the equation and solve for the coefficients a, b, and c.

Plugging in (1, 0):
0 = a(1)^2 + b(1) + c
0 = a + b + c

Plugging in (2, 5):
5 = a(2)^2 + b(2) + c
5 = 4a + 2b + c

Plugging in (4, 21):
21 = a(4)^2 + b(4) + c
21 = 16a + 4b + c

Now we have three equations with three unknowns (a, b, and c). We can solve this system of equations to find the values of a, b, and c, and thus determine the quadratic function.

4.) To determine the concavity of a parabola, you need to examine the coefficient of the x^2 term in the quadratic function. If it is positive, the parabola opens upward and is concave up. If it is negative, the parabola opens downward and is concave down.

For the given function f(x) = -2x2 - 3x + 35, the coefficient of the x^2 term is -2, which is negative. Therefore, the parabola is concave down.

5.) To find the maximum height of the projectile, you need to determine the vertex of the quadratic function h(t) = -16t^2 + 96t.

This function is already in the form f(t) = at^2 + bt + c, where a = -16, b = 96, and c = 0.

The vertex is given by the formula h = -b/(2a), where t represents the time.

Plugging in the values, we have:

t = -96 / (2 * -16)
t = -96 / -32
t = 3

Substituting t back into the equation to obtain the maximum height:

h(3) = -16(3)^2 + 96(3)
h(3) = -144 + 288
h(3) = 144

Therefore, the maximum height of the projectile is 144 feet.