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 the equation g(x)=x÷2-7÷2
1.1 Show that the coordinates of C are (7;0).
If y = x/2 - 7/2
and it passes through C and E which are the x and y intercepts respectively, the
for C, let y = 0
0 = x/2 - 7/2
x/2 = 7/2
x = 7,
so C is (7,0)
Why all that extra stuff about parabolas etc ?
Calculate x -coordinates of B
To show that the coordinates of C are (7,0), we need to find the x-intercept of the parabola.
Since the axis of symmetry of the parabola is x = 3, it means that the parabola is symmetric with respect to the line x = 3. This implies that the x-coordinate of the vertex of the parabola is 3.
The vertex of a parabola lies on the axis of symmetry, so its x-coordinate is also the average of the x-intercepts B and C.
Given that the equation of the axis of symmetry is x = 3, we can conclude that x = 3 is the x-coordinate of both B and C.
We know that the line g(x) passes through E and C. Since E lies on the y-axis, its x-coordinate is 0. Let's substitute x = 0 into the equation of the line g(x) to find the y-coordinate of E.
g(x) = x/2 - 7/2
g(0) = (0)/2 - 7/2 = -7/2
So, the coordinates of E are (0, -7/2).
Since C has the same x-coordinate as E, we can conclude that the coordinates of C are (0, -7/2).
To determine the x-coordinate of C, we know that the parabola intersects the x-axis at B and C. Since B and C have the same x-coordinate, we can conclude that the x-intercept of the parabola is x = 0.
Therefore, the coordinates of C are (0, 0).
However, this contradicts the given information that the parabola intersects the x-axis at C.
Hence, there seems to be an error in the problem statement or in the previous steps of the solution.
To show that the coordinates of point C are (7,0), we need to find the x-coordinate of point C where the parabola intersects the x-axis.
Since the axis of symmetry of the parabola has equation x = 3, it means that the vertex of the parabola lies on the line x = 3. The vertex of a parabola is the point where the parabola transitions from moving upward to moving downward, and it lies on the axis of symmetry.
Since the x-coordinate of the vertex is 3, we can plug this value into the equation of the line g(x) to find the y-coordinate of the vertex.
g(3) = 3/2 - 7/2
g(3) = -4/2
g(3) = -2
Therefore, the vertex of the parabola is (3, -2).
Since the parabola intersects the y-axis at point E, we know that the x-coordinate of point E is 0. To find the y-coordinate of point E, we can plug this value into the equation of the parabola.
Using the equation of the parabola, we have f(x) = a(x - h)^2 + k, where (h, k) denotes the coordinates of the vertex.
Substituting the values (3, -2) into the equation, we have f(x) = a(x - 3)^2 - 2.
Plugging in x = 0 to find the y-coordinate of point E, we have f(0) = a(0 - 3)^2 - 2.
f(0) = 9a - 2
Since point E lies on the y-axis, its x-coordinate is 0 and its y-coordinate is therefore f(0).
Since point E is on the y-axis, we know the x-coordinate of point C is 0. Therefore, we need to find the y-coordinate of point C.
Since point C lies on the parabola, we can use the same equation as above with unknown constant a, but substitute in x = 0.
Hence, f(0) = a(0 - 3)^2 - 2.
Now we can set these two equations equal to each other, since both equations represent the y-coordinate of point E and C respectively.
9a - 2 = a(0 - 3)^2 - 2
9a - 2 = 9a - 2
This equation is true for all values of a. It means that whatever value of a we choose, as long as it is not equal to zero, the equation holds. However, this doesn't give us any information about the specific value of a.
So, we cannot solve for the y-coordinate of point C directly with this information. We need additional information to find the specific coordinates of point C.