What is the y-intercept, the equation of the axis of symmetry, and the x-coordinates of the vertex?
Can you also make table of values that includes the vertex?
f(x) = 3x^2 + 8x
the vertex (and also the axis of symmetry) lies on the line x = -b/2a
That is, x = -8/6 = -4/3
The y=intercept is clearly at (0,0)
To find the y-intercept, we need to determine the value of the y-coordinate when x is equal to 0. So, we substitute x=0 into the equation:
f(0) = 3(0)^2 + 8(0) = 0
Therefore, the y-intercept is 0.
To find the equation of the axis of symmetry, we need to remember that for a quadratic equation in the form f(x) = ax^2 + bx + c, the equation of the axis of symmetry is given by x = -b/2a.
In this case, a = 3 and b = 8.
The equation of the axis of symmetry is:
x = -8 / (2 * 3) = -8 / 6 = -4/3
So, the equation of the axis of symmetry is x = -4/3.
To find the x-coordinate of the vertex, we use the same formula x = -b/2a.
x = -8 / (2 * 3) = -8 / 6 = -4/3
So, the x-coordinate of the vertex is -4/3.
To find the y-coordinate of the vertex, we substitute the x-coordinate of the vertex (-4/3) back into the original equation:
f(-4/3) = 3(-4/3)^2 + 8(-4/3) = 3(16/9) - 32/3 = 48/9 - 32/3 = 16/3 - 32/3 = -16/3
So, the y-coordinate of the vertex is -16/3.
Now, let's create a table of values including the vertex:
x | f(x)
---------
-2 | 4
-1 | -5
0 | 0
1 | 11
2 | 32
-4/3 | -16/3
3 | 57
4 | 92
In this table, we substituted different values for x into the equation f(x) = 3x^2 + 8x to find the corresponding values of f(x). The row where x = -4/3 represents the vertex point.