### Understanding Quadratic Functions#### Example Quadratic Function:The graph of a quadratic function $ f $ is given as:\[ f(x) = -\frac{1}{2}x^2 - 2x + 6 \]The function is represented by a parabola opening downwards. #### Graph Explanation:The graph shows a parabola that peaks at a certain point and intersects the $ x $-axis at two points and the $ y $-axis at one point. The highest point on the parabola is its vertex, and the points where it intersects the $ x $-axis are its roots or $ x $-intercepts. The point where it intersects the $ y $-axis is its $ y $-intercept.#### Parts of the Graph:- The vertex represents the maximum/minimum point of the function.- $ x $-intercepts are the points where the graph crosses the $ x $-axis.- $ y $-intercept is the point where the graph crosses the $ y $-axis.### Investigative Questions(a) **Finding the Coordinates of the Vertex, $ x $- and $ y $-Intercepts**:Fill in the coordinates by looking at the graph or solving the equation.- Vertex: $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_\_\_)- \( x $-Intercepts: - $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$ (smaller $ x $-value) - $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$ (larger $ x $-value)- $ y $-Intercept: $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$(b) **Finding the Maximum or Minimum Value of $ f $**:Determine whether the function has a maximum or minimum value and state what it is.- The $ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ $ value of $ f $ is $ f(x) = \_\_\_\_\_\_\_\_\_\_\_\_$.(c) **Domain and Range**:Identify the domain and range of the function using interval notation.- Domain: \_\_\_\_\_\_\

a). Given, y= f(x)= -0.5x²-2x+6 --------equation 1 Vertex is nothing but the maximum point of the…

The graph of a quadratic function f is given. f(x) = - - 2x + 6 y 10 WebAssign Plot X (a) Find the coordinates of the vertex and the x- and y-intercepts. vertex (х, у) %3D x-intercepts (х, у) %3D (smaller x-value) (х, у) %3D (larger x-value) y-intercept (х, у) %3D (b) Find the maximum or minimum value of f. The ---Select--- + value of f is f(x) = (c) Find the domain and range of f. (Enter your answers using interval notation.) domain rapge

The graph of a quadratic function f is given. f(x) = - - 2x + 6 y 10 WebAssign Plot X (a) Find the coordinates of the vertex and the x- and y-intercepts. vertex (х, у) %3D x-intercepts (х, у) %3D (smaller x-value) (х, у) %3D (larger x-value) y-intercept (х, у) %3D (b) Find the maximum or minimum value of f. The ---Select--- + value of f is f(x) = (c) Find the domain and range of f. (Enter your answers using interval notation.) domain rapge

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Concept explainers

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

Question

### Understanding Quadratic Functions

#### Example Quadratic Function:
The graph of a quadratic function $ f $ is given as:

\[ f(x) = -\frac{1}{2}x^2 - 2x + 6 \]

The function is represented by a parabola opening downwards.

#### Graph Explanation:
The graph shows a parabola that peaks at a certain point and intersects the $ x $-axis at two points and the $ y $-axis at one point. The highest point on the parabola is its vertex, and the points where it intersects the $ x $-axis are its roots or $ x $-intercepts. The point where it intersects the $ y $-axis is its $ y $-intercept.

#### Parts of the Graph:
- The vertex represents the maximum/minimum point of the function.
- $ x $-intercepts are the points where the graph crosses the $ x $-axis.
- $ y $-intercept is the point where the graph crosses the $ y $-axis.

### Investigative Questions
(a) **Finding the Coordinates of the Vertex, $ x $- and $ y $-Intercepts**:
Fill in the coordinates by looking at the graph or solving the equation.

- Vertex: $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_\_\_)
- \( x $-Intercepts:
- $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$ (smaller $ x $-value)
- $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$ (larger $ x $-value)
- $ y $-Intercept: $(x, y) = \_\_\_\_\_\_\_\_\_\_\_\_$

(b) **Finding the Maximum or Minimum Value of $ f $**:
Determine whether the function has a maximum or minimum value and state what it is.

- The $ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ $ value of $ f $ is $ f(x) = \_\_\_\_\_\_\_\_\_\_\_\_$.

(c) **Domain and Range**:
Identify the domain and range of the function using interval notation.

- Domain: \_\_\_\_\_\_\

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