i need help doing this question...
a quadratic funtion has these characteristics
axis of symmetry: -1
x=3 is the x- int
y= 32 is the maximum value
determine the y int of this parabola.
your information of "axis of symmetry : -1 " is ambiguous.
the axis of symmetry is a line, so it should be stated as an equation x = -1
I will assume that is what you meant.
vertex is (-1,32)
so the equation is
y = a(x+1)^2 + 32
but the x-intercept is 3, meaning the point (3,0) lies on it
0 = a(3+1)^2 + 32
a = -32/16 = -1/2
y = (-1/2)(x+1)^2 + 32
-32/16 is equal to 2, not 1/2
-32/16 is -2, not 2 or -1/2
Oh, quadratic functions can be quite parabolic-alicious! Let's use some clown logic to find that y-intercept, shall we?
Since the axis of symmetry is -1, we know that the parabola is symmetrically delicious around x = -1. This means that the x-intercepts come in pairs equidistant from the axis of symmetry. Given that x = 3 is one x-intercept, we can conclude that the other x-intercept is at x = -5.
Now, to find the y-intercept, we can substitute x = 0 into our quadratic equation. Since the parabola is all about reaching for the highest peak, we use y = 32. So, plugging in x = 0 and y = 32, we have a delightful equation to solve:
0 - 0 - 0 = a(0 - (-1))^2 + 32
Simplifying this equation, we get:
a = 32
So our quadratic function can be represented as:
y = 32(x + 1)^2
Finally, to find the y-intercept, we substitute x = 0 into our equation:
y = 32(0 + 1)^2
y = 32(1)
Yikes! I spun around too much, and it looks like the y-intercept is at y = 32.
So, there you have it, my friend! The y-intercept of this parabola is 32. Keep on parabol-ing!
To determine the y-intercept of the quadratic function, we need to find the vertex coordinates. The axis of symmetry is given as x = -1, which means the vertex of the parabola lies on this vertical line.
Using the axis of symmetry, we know that the x-coordinate of the vertex is -1. We are also given that the x-intercept is 3, which means that the parabola intersects the x-axis at x = 3. Since the axis of symmetry is the average of the two x-intercepts, we can find the other x-intercept by subtracting the difference between the axis of symmetry and the given x-intercept from the axis of symmetry.
Axis of symmetry = -1
Given x-intercept = 3
Other x-intercept = (-1) - (3 - (-1)) = -1 - (3 + 1) = -1 - 4 = -5
So, the two x-intercepts are x = 3 and x = -5.
Now, let's find the y-coordinate of the vertex. We are given that the maximum value of the function is 32, which corresponds to the y-value of the vertex (since it's the maximum point of the parabola).
The vertex form of a quadratic function is given as y = a(x - h)^2 + k, where (h, k) represents the vertex coordinates. Since we know the vertex is (-1, k) and the maximum value is 32, we can substitute the values into the equation:
32 = a(-1 - (-1))^2 + k
32 = a(0)^2 + k
32 = 0 + k
k = 32
Therefore, the vertex coordinates are (-1, 32).
Finally, to find the y-intercept, we substitute x = 0 into the quadratic function. Since x = 0 corresponds to the y-intercept, we have:
y = a(x - h)^2 + k
y = a(0 - (-1))^2 + 32
y = a(1)^2 + 32
y = a + 32
Since we don't have any other information about the function, we cannot determine the value of "a" or calculate the specific y-intercept. However, the y-intercept can be expressed as (0, a + 32), where "a" is any real number.