Determine the x intercepts, y intercept, equation of the axis of symmetry and vertex for each of the following functions,
1.) f(x)= 3(x-2)^2-6 with zeros of 0.59 and 3.41
2.) g(x)= (x-4)(x+7)
1. f(x)= 3(x-2)^2-6 with zeros of 0.59 and 3.41
everything you asked for , except the y-intercept, can simply be stated directly from the equation:
x-intercepts, you stated them already
axis of symmetry: x-2 = 0 or x = 2
vertex: (2,-6)
for the y-intercept, let x = 0
y = 3(-2)^2 - 6 = 6
g(x) = (x-4)(x+7)
x intercepts: x = 4 and x = -7
the vertex is half-way between 4 and -7
so the x of the vertex is (-7+4)/2 = -3/2
sub that in ...
y = (-3/2 - 4)(-3/2 + 7) = -121/4
vertex is (-3/2 , -121/4)
axis of symmetry : x = -3/2
yintercept: let x = 0
y = (-4)(7) = -28
To determine the x-intercepts, y-intercept, equation of the axis of symmetry, and vertex for each function, let's solve them step by step.
1) For the function f(x) = 3(x-2)^2 - 6:
a) X-intercepts:
To find the x-intercepts, we need to set f(x) equal to zero and solve for x.
Setting f(x) = 0:
3(x-2)^2 - 6 = 0
Dividing both sides by 3:
(x-2)^2 - 2 = 0
Expanding the square:
(x^2 - 4x + 4) - 2 = 0
Simplifying:
x^2 - 4x + 4 - 2 = 0
x^2 - 4x + 2 = 0
Next, we can use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -4, and c = 2. Plugging in the values, we get:
x = (-(-4) ± √((-4)^2 - 4(1)(2))) / (2*1)
x = (4 ± √(16 - 8)) / 2
x = (4 ± √8) / 2
x = (4 ± 2√2) / 2
Simplifying further:
x = 2 ± √2
Therefore, the x-intercepts are approximately x = 2 + √2 and x = 2 - √2, or in decimal form, x ≈ 3.41 and x ≈ 0.59.
b) Y-intercept:
To find the y-intercept, we need to evaluate f(0).
Plugging in x = 0 into the function f(x):
f(0) = 3(0-2)^2 - 6
f(0) = 3(-2)^2 - 6
f(0) = 3(4) - 6
f(0) = 12 - 6
f(0) = 6
Therefore, the y-intercept is y = 6.
c) Equation of the axis of symmetry:
The equation of the axis of symmetry is given by the formula x = -b/2a.
In this case, a = 3 and b = 0 (since there is no x-term in the equation).
Substituting the values, we get:
x = -0 / (2*3)
x = 0 / 6
x = 0
Therefore, the equation of the axis of symmetry is x = 0.
d) Vertex:
The vertex can be found using the x-coordinate obtained in the equation of the axis of symmetry and by plugging it back into the original function f(x).
Substituting x = 0 into the equation f(x) = 3(x-2)^2 - 6:
f(0) = 3(0-2)^2 - 6
f(0) = 3(-2)^2 - 6
f(0) = 3(4) - 6
f(0) = 12 - 6
f(0) = 6
So, the vertex is (0, 6).
2) For the function g(x) = (x-4)(x+7):
a) X-intercepts:
To find the x-intercepts, we need to set g(x) equal to zero and solve for x.
Setting g(x) = 0:
(x-4)(x+7) = 0
Using the Zero Product Property, we set each factor equal to zero:
x-4 = 0 and x+7 = 0
Solving each equation individually, we have:
x = 4 and x = -7
Therefore, the x-intercepts are x = 4 and x = -7.
b) Y-intercept:
To find the y-intercept, we need to evaluate g(0).
Plugging in x = 0 into the function g(x):
g(0) = (0-4)(0+7)
g(0) = (-4)(7)
g(0) = -28
Therefore, the y-intercept is y = -28.
c) Equation of the axis of symmetry:
The equation of the axis of symmetry is not applicable for this function since it is a quadratic equation in the form (x - h)^2.
d) Vertex:
To find the vertex, we need to find the x-coordinate of the vertex by using the axis of symmetry formula (x = -b/2a) and then substitute it back into the original function g(x).
Since there is no x-term (b = 0) in the equation, the axis of symmetry is x = 0.
Substituting x = 0 into the function g(x):
g(0) = (0-4)(0+7)
g(0) = (-4)(7)
g(0) = -28
So, the vertex is (0, -28).
To determine the x-intercepts, we need to find the values of x when y equals zero in the given functions.
1.) For the function f(x) = 3(x-2)^2 - 6:
To find the x-intercepts, set f(x) equal to zero:
0 = 3(x-2)^2 - 6
Now, let's solve for x:
3(x-2)^2 - 6 = 0
Divide both sides by 3:
(x-2)^2 - 2 = 0
Expanding the equation:
x^2 - 4x + 4 - 2 = 0
Simplifying:
x^2 - 4x + 2 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -4, c = 2.
x = (-(-4) ± √((-4)^2 - 4(1)(2))) / (2(1))
x = (4 ± √(16 - 8)) / 2
x = (4 ± √8) / 2
x = (4 ± 2√2) / 2
x = 2 ± √2
Therefore, the x-intercepts for the function are x = 2 + √2 and x = 2 - √2.
To find the y-intercept, substitute x = 0 into the equation:
f(0) = 3(0-2)^2 - 6
f(0) = 3(-2)^2 - 6
f(0) = 3(4) - 6
f(0) = 12 - 6
f(0) = 6
Therefore, the y-intercept is y = 6.
The equation of the axis of symmetry is given by x = h, where h is the x-coordinate of the vertex. In this case, the vertex form of the equation is f(x) = a(x-h)^2 + k.
Comparing the given function to the vertex form, we can identify that h = 2.
So, the equation of the axis of symmetry is x = 2.
Lastly, to find the vertex, we use the formula x = h and substitute it into the function:
f(2) = 3(2-2)^2 - 6
f(2) = 3(0)^2 - 6
f(2) = 3(0) - 6
f(2) = -6
Therefore, the vertex is (2, -6).
2.) For the function g(x) = (x-4)(x+7):
To find the x-intercepts, set g(x) equal to zero:
0 = (x-4)(x+7)
Now, let's solve for x:
x-4 = 0 or x+7 = 0
x = 4 or x = -7
Therefore, the x-intercepts for the function are x = 4 and x = -7.
To find the y-intercept, substitute x = 0 into the equation:
g(0) = (0-4)(0+7)
g(0) = (-4)(7)
g(0) = -28
Therefore, the y-intercept is y = -28.
Since the given function is in factored form, the equation of the axis of symmetry can be determined by finding the average of the x-intercepts.
x = (4 + (-7)) / 2
x = -3/2
So, the equation of the axis of symmetry is x = -3/2.
Since the given function is a quadratic with real roots, the axis of symmetry passes through the vertex. Therefore, the vertex is located on the axis of symmetry, x = -3/2.
To find the y-coordinate of the vertex, substitute x = -3/2 into the equation:
g(-3/2) = (-3/2 - 4)(-3/2 + 7)
g(-3/2) = (-11/2)(11/2 - 3/2)
g(-3/2) = (-11/2)(8/2)
g(-3/2) = (-11/2)(4)
g(-3/2) = -22
Therefore, the vertex is (-3/2, -22).