Let's break down the key parts of the parabola based on the information provided:
-
Vertex: The vertex is given as (3, -9).
Response: (3, -9) -
Lesser Zero: The lesser zero (or root) is given as (0, 0).
Response: (0, 0) -
Other Zero: The other zero (or root) is given as (6, 0).
Response: (6, 0) -
Line of Symmetry: The line of symmetry is given as x = 3.
Response: x = 3 -
Y-Intercept: The y-intercept should typically be a point where the parabola crosses the y-axis, often found by evaluating the function at x = 0. The point is given as (0, 0) above, so it seems consistent.
Response: (0, 0) -
Complete the table below: To fill out the table, we first need to find the function (f(x)) that describes the parabola.
Given:
- The vertex is (3, -9).
- The roots are (0, 0) and (6, 0).
First, we can express the parabola in vertex form and then standard form or use the roots to establish a quadratic function.
Quadratic Function: The parabola can be expressed as: \[ f(x) = a(x - 0)(x - 6) \] This simplifies to: \[ f(x) = a(x^2 - 6x) \]
Since we also have the vertex (3, -9), we can use it to find 'a'.
Substituting \(x = 3\) into the equation: \[ f(3) = a(3 - 0)(3 - 6) = -9 \] \[ f(3) = a(3)(-3) = -9 \ -9a = -9 \ a = 1 \]
Thus, the equation is: \[ f(x) = (x - 0)(x - 6) = x^2 - 6x \]
Now we can calculate f(x) for the specified x values:
-
For x = -1: \( f(-1) = (-1)^2 - 6(-1) = 1 + 6 = 7 \)
-
For x = 0: \( f(0) = 0^2 - 6(0) = 0 \)
-
For x = 1: \( f(1) = (1)^2 - 6(1) = 1 - 6 = -5 \)
-
For x = 3: \( f(3) = (3)^2 - 6(3) = 9 - 18 = -9 \)
-
For x = 5: \( f(5) = (5)^2 - 6(5) = 25 - 30 = -5 \)
Putting it all together, the completed table is:
| x | f(x) | |----|------| | -1 | 7 | | 0 | 0 | | 1 | -5 | | 3 | -9 | | 5 | -5 |
So, the final answers are:
- (3, -9)
- (0, 0)
- (6, 0)
- x = 3
- (0, 0)
| x | f(x) | |----|------| | -1 | 7 | | 0 | 0 | | 1 | -5 | | 3 | -9 | | 5 | -5 |
4 answers