Use the function to answer these questions

Question

Function- y=-3x²+8x+35

 

identify the axis of symmetry and the vertex

 

identify the maximum

 

find the x-intercepts by factoring

 

show the value of the discriminant supports your conclusion

 

describe the relationship of the x-intercepts to the axis of symmetry

Answer 1

Equation: y=-3x²+8x+35

 

Axis of symmetry:

The axis of symmetry is the same as the x value of the vertex since vertex occurs at the middle of the parabola and that is where the symmetrical line is. To find it, we can use the formula h = -b / 2a. Remember the equation y=-3x²+8x+35 is in standard form (y=ax²+bx+c) so we know that b is 8 and a is -3 so we can plug that into the formula:

h = -8 / 2(-3) = -8 / -6 = 4 / 3

Since axis of symmetry is a line, it would be an equation x = 4 / 3

Vertex:

We just found the x value of the vertex which was the same as the axis of symmetry, now to find the y point of the vertex, we can plug in our x point into the equation:

y = -3(4/3)² + 8(4/3) + 35

= -3(16/9) + 32/3 + 35

= -48/9 + 32/3 + 35

= -48/9 + 96/9 + 315/9

= 363/9

= 121/3

Vertex: (4/3, 121/3)

Maximum:

The maximum would be the y coordinate of the vertex because that is where the graph reaches its maximum.

Maximum: 121/3

x-intercepts:

In order to find the x-intercepts, we can factor our quadratic function. To factor the equation y= -3x²+8x+35, we can apply the ac method. In the ac method, we first multiply a by c so -3 * 35 = -105. Then we have to find two numbers that are a product (multiply up to) of -105 and at the same time add up to 8 (the b in the equation). Those two numbers would be -7 and 15. The next step is to divide each of those by a so -7/-3 and 15/-3. We simply those two fractions: -7/-3 stays the same since it can't be simplified further and 15/-3 is simplified down to -5/1. Now we can take the denominator of the first fraction and write it with the x and then put the numerator along with it and then do the same with the second fraction which will give us our factored form:

(-3x - 7)(x - 5) = 0

To find the x-intercept now, we can use the zero product rule to equal each factor to 0 and then solve for x:

-3x - 7 = 0 x - 5 = 0

+7 +7 +5 +5

-------------- -------------

-3x = 7 x = 5

x = -7/3

x-intercepts: (-7/3, 0) and (5, 0)

Discriminant:

The discriminant can be found by the discriminant formula: D = b² - 4ac. So, we can plug in out a, b and c into the formula:

D = 8² - 4(-3)(35)

= 64 + 420

= 484

Discriminant = 484 which is greater than 0 which indicates that it has two real solutions and two distinct x-intercepts and this matches what we found by factoring. We found two real solutions that were distinct from each other.

Relationship of the x-intercepts to the axis of symmetry:

The axis of symmetry is always the midpoint of the x-intercepts. To confirm that, we can use the midpoint formula to find the middle value of the x-intercept and that should be equivalent to the axis of symmetry:

Midpoint formula: ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

= ((-7/3 + 5) / 2, (0 + 0) / 2)

= ((-7/3 + 15/3) / 2, 0)

= (4/3, 0)

We got (4/3, 0) as our midpoint for the x-intercepts and that was the same as our axis of symmetry. Hence, it proved that the axis of symmetry lies exactly halfway between the x-intercepts. So, the x-intercepts are equally spaced on both sides of the axis.

Answer 2

 

y= -3x^2+8x+35

  = -3(x^2-8x-35)

   = -3(x-4/3)^2+121/3

 

So,vertex=(4/3,121/3)

Axis of symmetry =.   x= 4/3

Maximum point =. y = 121/3

 

x intercepts by factoring

 

y =-3x^2+8x+35

 

applying Discriminant formula

 

y= (-8+- sqrt(64+420))/-6

 

   (-8+22)/-6     &.  (-8-22)/-6

 

   -7/3.  &.  5

 

so  x  axis intercepts.  are. -7/3. &  5

 

 since axis of symmetry is x= 4/3

 

7/3+4/3  11/3

 

5-4/3=11/3

 

So, this justifies the x axis intercepts.