The graph of a quadratic function f is given. f(x) = -x2 + 8x -7 y 1 (a) Find the coordinates of the vertex and the x- and y-intercepts. vertex (x, y) = x-intercepts (x, y) = (smaller x-value) (x, y) = (larger x-value) y-intercept (x, y) = (b) Find the maximum or minimum value of f. The -Select--V value of f is f(x) = (c) Find the domain and range of f. (Enter your answers using interval notation.) domain range
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Quadratic Functions: Finding Key Features
#### The graph of a quadratic function \( f \) is given.
\[ f(x) = -x^2 + 8x - 7 \]
*Explanation of the Graph:*
The given quadratic function \( f(x) = -x^2 + 8x - 7 \) is represented by a downward-facing parabola on the coordinate plane. The vertex of the parabola is a point where the graph reaches its maximum value since the parabola opens downwards. The x- and y-intercepts are the points where the graph intersects the x-axis and y-axis, respectively.
**(a) Find the coordinates of the vertex and the x- and y-intercepts.**
- **Vertex**: The vertex of the quadratic function is the highest or lowest point of the parabola. For the function in standard form \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be found using \( x = -\frac{b}{2a} \).
\[
(x, y) = \left( [ \ ], [ \ ] \right)
\]
- **x-intercepts**: The points where the graph intersects the x-axis. Solve the equation \( -x^2 + 8x - 7 = 0 \) to find the x-intercepts.
\[
(x, y) = \left( [ \ ], 0 \right) \quad \text{(smaller x-value)}
\]
\[
(x, y) = \left( [ \ ], 0 \right) \quad \text{(larger x-value)}
\]
- **y-intercept**: The point where the graph intersects the y-axis. This can be found by evaluating the function at \( x = 0 \).
\[
(x, y) = \left( 0, [ \ ] \right)
\]
**(b) Find the maximum or minimum value of \( f \).**
Since the parabola opens downward, the function \( f \) has a maximum value at the vertex.
The [Select] value of \( f \) is \( f(x) = [ \] \).
**(c) Find the domain and range of \( f \).** (Enter your answers](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f4719ec-89c6-49f2-a8fd-c905fe776b01%2F2a81ee53-6eec-4ee8-8a9f-1eefa4c230ed%2Fsa62t0o_processed.jpeg&w=3840&q=75)
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