For each of the following equations.
Find the coordinates of the vertex of the curve it describes.
Find the x− intercepts.
Find the y− intercept.
Find the equation of the line of symmetry.
Use all this information to sketch the graph of the function.
y =2−(x −3)^2
This is just a parabola opening downward.
The given equation is in vertex form, so just read off the vertex from that.
when x=0, y=?
When y=0, x-3 = ±√2
The line of symmetry goes through the vertex.
thanks
To find the coordinates of the vertex of the given equation, we first need to identify the values of "h" and "k" in the general vertex form equation, which is y = a(x - h)^2 + k.
Comparing this with the given equation, y = 2 - (x - 3)^2, we can see that "h" = 3 and "k" = 2. Therefore, the vertex coordinates are (3, 2).
To find the x-intercepts, we substitute y = 0 into the given equation and solve for x. So, we have 0 = 2 - (x - 3)^2. Expanding and rearranging the equation, we get (x - 3)^2 = 2. Taking the square root of both sides, we have x - 3 = ±√2. Therefore, x = 3 + √2 and x = 3 - √2 are the x-intercepts.
To find the y-intercept, we substitute x = 0 into the given equation. Thus, we have y = 2 - (0 - 3)^2 = 2 - 9 = -7. Therefore, the y-intercept is (0, -7).
The equation of the line of symmetry for this quadratic function is x = h, where "h" is the x-coordinate of the vertex. In this case, the line of symmetry is x = 3.
To sketch the graph of the function, we plot the vertex at (3, 2) and the x-intercepts at (3 + √2, 0) and (3 - √2, 0). We also plot the y-intercept at (0, -7). Then we use the line of symmetry to mirror the points to the other side. Finally, we draw a U-shaped curve through these points to complete the graph.