Answer:
the point (5/2, -27/2) is also on the graph of the function. By plotting these points, we can sketch the graph of the function.
Explanation:
To find the x-intercepts of the graph of f(x) = 2x^2 – 9x + 9, we set f(x) equal to zero and solve for x:
2x^2 – 9x + 9 = 0
We can use the quadratic formula to solve for x:
x = [-(b) ± sqrt((b^2) - 4(a)(c))] / 2(a)
Plugging in the values from our equation, we get:
x = [-( -9 ) ± sqrt( (-9)^2 - 4(2)(9) )] / 2(2)
x = [9 ± sqrt(9)] / 4
x = (9 ± 3) / 4
x = 3/2, 3
Therefore, the x-intercepts of the graph are (3/2, 0) and (3, 0).
To find the coordinates of the vertex, we use the formula:
x = -b / 2a
We know that a = 2 and b = -9 from our equation, so we get:
x = -(-9) / 2(2)
x = 9/4
To find the corresponding y-coordinate of the vertex, we plug this value of x back into our equation:
f(9/4) = 2(9/4)^2 – 9(9/4) + 9
f(9/4) = 81/8 - 81/4 + 9
f(9/4) = -27/8
Therefore, the vertex of the graph is (9/4, -27/8).
To graph this function, we can plot the vertex at (9/4, -27/8) and the x-intercepts at (3/2, 0) and (3, 0). We can also plot a point on either side of the vertex by choosing an x-value and finding the corresponding y-value using the equation. For example, if we plug in x = 2, we get:
f(2) = 2(2)^2 - 9(2) + 9
f(2) = -7
So the point (2, -7) is on the graph of the function. Similarly, if we plug in x = 5/2, we get:
f(5/2) = 2(5/2)^2 - 9(5/2) + 9
f(5/2) = -27/2
So the point (5/2, -27/2) is also on the graph of the function. By plotting these points, we can sketch the graph of the function.