Answer:
Standard form: f(x) = -x^2 + 6x + 16
Explanation:
Identifying the function:
- The graph is a parabola, which means f(x) is a quadratic function.
General equation of a quadratic's standard form:
The general equation for the standard form of a quadratic function is given by:
f(x) = ax^2 + bx + c
Method to find the standard form of the equation of f(x):
We can find either the vertex form or the intercept form, whose general equations are given by:
Vertex form: f(x) = a(x - h)^2 + k, where
- a is a constant determining whether the parabola opens upward or downward,
- and (h, k) are the coordinates of the vertex.
Intercept form: f(x) = a(x - p)(x - q), where
- a is a constant determining whether the parabola opens upward or downward,
- and p and q are the x-intercepts (aka roots).
Connection between the vertex and intercepts forms and the standard form:
- As both these forms show, we're able to expand and expanding and simplifying yields the standard form.
- Let's convert from vertex form to standard form.
Identifying the vertex:
- This quadratic opens downward, which means the vertex is the highest point on the graph.
- The coordinates of the vertex are (3, 25)
Finding the vertex form:
- We use the opposite sign of the x-coordinate when plugging in the vertex into the vertex form
Thus, we have part of the vertex form: f(x) = a(x - 3)^2 + 25.
- We can find a by plugging in a point that lies on the parabola for (x, f(x)) in the vertex form.
- We see that the point (-2, 0) iies on the parabola as it is one of the x-intercepts.
Thus, we can plug in (-2, 0) for (x, f(x)) in the vertex form to find a:
0 = a(-2 - 3)^2 + 25
0 = a(-5)^2 + 25
0 = a * 25 + 25
(0 = 25a + 25) - 25
(-25 = 25a) / 25
-1 = a
Thus, -1 is a in the vertex form.
Therefore, the full vertex form is given by:
f(x) = -(x - 3)^2 + 25
Convert from vertex form the standard form:
Now we can convert to standard for by expanding:
f(x) = -(x - 3)(x - 3) + 25
f(x) =-(x^2 - 3x - 3x + 9) + 25
f(x) = -x^2 + 3x + 3x - 9 + 25
f(x) = -x^2 + 6x + 16
Thus, the standard form of the equation of f(x) is f(x) = -x^2 + 6x + 16