For the graph y find both x-intercepts by solving the equation 3x^2 - 24x + 45 =0
a) Direction of opening
b)y-intercepts
c)Axis of symmetry
d) vertex
e) max or min
y = 3(x-3)(x-5) = 3(x-4)^2 - 3
now the answers should be clear.
What do you get?
To find the x-intercepts of the graph of the equation 3x^2 - 24x + 45 = 0, we need to solve the equation for when y is equal to zero.
a) Direction of opening:
The leading coefficient of the quadratic equation is 3, which is positive, so the parabola opens upward.
b) y-intercepts:
To find the y-intercept, we substitute x = 0 into the equation.
3(0)^2 - 24(0) + 45 = 45
Therefore, the y-intercept is (0, 45).
c) Axis of symmetry:
The equation of the axis of symmetry is given by x = -b/2a, where a, b, and c are coefficients in the quadratic equation ax^2 + bx + c = 0.
In our equation, a = 3 and b = -24.
x = -(-24) / (2 * 3) = 24 / 6 = 4
Therefore, the axis of symmetry is x = 4.
d) Vertex:
The x-coordinate of the vertex is the same as the axis of symmetry, which is x = 4.
To find the y-coordinate of the vertex, we substitute x = 4 into the equation.
3(4)^2 - 24(4) + 45 = 3(16) - 96 + 45 = 48 - 96 + 45 = -3
Therefore, the vertex is (4, -3).
e) Max or min:
Since the parabola opens upward, the vertex represents the minimum point on the graph. Therefore, the graph has a minimum value at the vertex.
To summarize:
a) The direction of opening is upward.
b) The y-intercept is (0, 45).
c) The axis of symmetry is x = 4.
d) The vertex is located at (4, -3).
e) The graph has a minimum at the vertex.