A parabola f intersect the x-
axis at B and C and the y-axis at
E.The axis of symmetry of the
parabola has equation x=3.The line
through E and C has equation g(x)=x/2-7/2. Calculate the x coordinate of B
Since the axis of symmetry is x=3 and the y-intercept is E, the parabola is
f(x) = a(x-3)^2 + E
since E is the y-intercept of the parabola, E = -7/2
f(x) = a(x-3)^2 - 7/2
Now you know that (7,0) is on the parabola, so
a(7-3)^2 - 7/2 = 0
16a = 7/2
a = 7/32
f(x) = 7/32 (x-3)^2 - 7/2
or, since you now know that the roots are at x= 7 and -1 (why?)
f(x) = 7/32 (x-7)(x+1)
Why did the parabola go to the circus? Because it wanted to find its X-axis! Let's help it out.
Since the axis of symmetry of the parabola has an equation x = 3, that means it's a vertical line passing through the vertex. We know that the x-intercepts are at B and C, so these points lie on the parabola.
Since the parabola intersects the x-axis at B and C, it means that their y-coordinates are zero. So, we have B(?, 0) and C(?, 0).
Now, let's look at the y-intercept E. Since it intersects the y-axis at E, its x-coordinate is 0. Therefore, we have E(0, ?).
Lastly, we're given the equation for the line passing through E and C, which is g(x) = x/2 - 7/2.
To calculate the x-coordinate of B, we need to find the point of intersection between the parabola and the line g(x).
Since B lies on the parabola and g(x) passes through C and E, we can equate their y-values and solve for the x-coordinate of B.
Setting the y-values equal to each other:
0 = x/2 - 7/2
To eliminate the fraction, let's multiply everything by 2:
0 = x - 7
Adding 7 to both sides:
7 = x
So the x-coordinate of B is 7.
Therefore, B(7, 0) is the point of intersection between the parabola and the x-axis.
To find the x-coordinate of point B, we need to determine the equation of the parabola. Let's break down the given information:
1. The axis of symmetry of the parabola has equation x = 3.
2. The parabola intersects the x-axis at points B and C.
3. The parabola intersects the y-axis at point E.
4. The line through E and C has equation g(x) = x/2 - 7/2.
Since the axis of symmetry is x = 3, the vertex of the parabola is located on this line. Therefore, the x-coordinate of the vertex is 3.
Let's consider the point of intersection between the line g(x) and the parabola. The x-coordinate of this point will be the same for both equations.
To find this x-coordinate, we can set g(x) equal to the x-coordinate of the vertex (3) and solve for x:
x/2 - 7/2 = 3
To simplify, let's multiply both sides by 2:
x - 7 = 6
Adding 7 to both sides:
x = 6 + 7
x = 13
Thus, the x-coordinate of the point of intersection between the line g(x) and the parabola is 13.
Since the parabola intersects the x-axis at points B and C, the x-coordinate of B is the same as the x-coordinate of the point of intersection, which is:
x-coordinate of B = 13
Therefore, the x-coordinate of point B is 13.
To find the x-coordinate of point B, where the parabola intersects the x-axis, we need to determine the point of intersection between the parabola and the line g(x). Here's how we can do that:
1. Equation of the Axis of Symmetry:
The given equation x = 3 represents the equation of the axis of symmetry of the parabola. The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. In this case, the line is vertical and intersects the x-axis at x = 3.
2. Vertex of the Parabola:
Since the axis of symmetry passes through the vertex of the parabola and the equation of the axis of symmetry is x = 3, the vertex of the parabola is at the point (3, y), where y is the y-coordinate of the vertex.
3. Equation of the Line g(x):
The equation of the line g(x) is given as g(x) = x/2 - 7/2.
4. Point of Intersection between g(x) and the Parabola:
To find the point of intersection between g(x) and the parabola, we need to find the x-coordinate(s) where g(x) = 0 (i.e., the line intersects the x-axis). Setting g(x) = 0, we have:
x/2 - 7/2 = 0
Solving this equation for x will give us the x-coordinate(s) of the intersection point(s).
5. Calculating the x-coordinate of B:
Now that we have the x-coordinate(s) of the intersection point(s) between g(x) and the parabola, we can determine the x-coordinate of point B. Point B represents the x-coordinate of the intersection between the parabola and the x-axis.
So, by following these steps, we can calculate the x-coordinate of point B.