### Understanding Functions through Table of Values and Graphs**Given:**The table of values represents the function $ g(x) $, while the graph shows the function $ f(x) $.#### Table of Values| $ x $ | $ g(x) $ ||-------|---------|| -2 | -3 || 1 | 0 || 2 | 1 |#### Question:Which statement about the functions is true?**Options:**A) The x-intercepts of $ f(x) $ are common to those of $ g(x) $.B) The minimum value of $ f(x) $ is less than the minimum value of $ g(x) $.C) $ f(x) $ and $ g(x) $ have the same y-intercept.D) $ f(x) $ and $ g(x) $ intersect at exactly two points.### Explanation:- **Option A)** - This statement suggests that the x-intercepts (where the function value is 0) of $ f(x) $ and $ g(x) $ are the same. - **Option B)** - This statement compares the minimum values of $ f(x) $ and $ g(x) $, indicating that $ f(x) $ reaches a lower minimum value.- **Option C)** - This statement implies that both functions intersect the y-axis at the same point, meaning they have the same y-intercept. - **Option D)** - This statement suggests that the functions $ f(x) $ and $ g(x) $ intersect at exactly two points on the graph.### Analysis:- From the table, the minimum value of $ g(x) $ is -3 at $ x = -2 $. - The $ g(x) $ crosses the x-axis (where $ g(x) = 0 $) at $ x = 1 $. - Further information is needed about $ f(x) $ and its graph to validate these statements fully.This exercise integrates understanding from numerical tables and graphical representations, encouraging comprehensive analytical skills in studying functions. ### Graph of a Quadratic Function#### Description:The graph shown is a quadratic function displayed on a Cartesian plane with both x- and y-axis ranges from -10 to 10. It is a U-shaped curve opening upwards, which indicates it is a parabola. The vertex of the parabola appears to be at the point $(-2, 1)$, making this the minimum point of the function. #### Detailed Features:1. **Axes:** - The horizontal axis (x-axis) and vertical axis (y-axis) intersect at the origin point (0, 0). - The x-axis and y-axis are labeled with increments of 1 unit, covering the range from -10 to 10.2. **Parabola:** - **Vertex:** The vertex, which is the lowest point of the parabola, is at $(-2, 1)$. - **Symmetry:** The parabola is symmetric with respect to the vertical line passing through its vertex. This means that if the x-coordinates of points are equidistant from $-2$ on the horizontal axis, their y-coordinates will be the same. - **Direction:** The parabolic curve opens upwards, implying that the coefficient of the $x^2$ term in its quadratic equation is positive.3. **Intersection Points with Axes:** - **Y-axis:** The parabola intersects the y-axis at $ (0, 4) $, indicating that when $ x = 0 $, $ y = 4 $. - **X-axis:** It intersects the x-axis at two points: approximately $ (-4, 0) $ and $ (0, 0) $. These are the roots or solutions of the quadratic equation when $ y = 0 $.#### Educational Context:This graph is an example of a quadratic function, which is fundamental in algebra and pre-calculus. Quadratic functions have the general form $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants. Understanding how to graph these functions and identify their key features, such as the vertex, axis of symmetry, and intercepts, is crucial for students studying quadratic relationships.#### Key Takeaways:- The graph of a quadratic function is always a parabola.- The direction in which the parabola opens (upwards or

First checking the Option A) Now the x-intercepts of f( x) where y=f(x)= 0 Form the graph of f(x) ,…

the table of values represents the funetion g (x) and the grayh Shows the function f (×) which statment about the funerions is trve? 9(x) -2 -3 the Y-intercupts of f (x) are common to those of g(x). 2. -3 R) the minimwm value of f (x) is less than the minimunm B) value of g(x). O ACX) and g(x) have the D Ganie y-intercepto 9(x). D) fX) and g x) intersect at uactly to points.

the table of values represents the funetion g (x) and the grayh Shows the function f (×) which statment about the funerions is trve? 9(x) -2 -3 the Y-intercupts of f (x) are common to those of g(x). 2. -3 R) the minimwm value of f (x) is less than the minimunm B) value of g(x). O ACX) and g(x) have the D Ganie y-intercepto 9(x). D) fX) and g x) intersect at uactly to points.

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

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.

Question

### Understanding Functions through Table of Values and Graphs

**Given:**
The table of values represents the function $ g(x) $, while the graph shows the function $ f(x) $.

#### Table of Values

| $ x $ | $ g(x) $ |
|-------|---------|
| -2 | -3 |
| 1 | 0 |
| 2 | 1 |

#### Question:
Which statement about the functions is true?

**Options:**
A) The x-intercepts of $ f(x) $ are common to those of $ g(x) $.

B) The minimum value of $ f(x) $ is less than the minimum value of $ g(x) $.

C) $ f(x) $ and $ g(x) $ have the same y-intercept.

D) $ f(x) $ and $ g(x) $ intersect at exactly two points.

### Explanation:

- **Option A)**
- This statement suggests that the x-intercepts (where the function value is 0) of $ f(x) $ and $ g(x) $ are the same.

- **Option B)**
- This statement compares the minimum values of $ f(x) $ and $ g(x) $, indicating that $ f(x) $ reaches a lower minimum value.

- **Option C)**
- This statement implies that both functions intersect the y-axis at the same point, meaning they have the same y-intercept.

- **Option D)**
- This statement suggests that the functions $ f(x) $ and $ g(x) $ intersect at exactly two points on the graph.

### Analysis:

- From the table, the minimum value of $ g(x) $ is -3 at $ x = -2 $.

- The $ g(x) $ crosses the x-axis (where $ g(x) = 0 $) at $ x = 1 $.

- Further information is needed about $ f(x) $ and its graph to validate these statements fully.

This exercise integrates understanding from numerical tables and graphical representations, encouraging comprehensive analytical skills in studying functions.

### Graph of a Quadratic Function

#### Description:
The graph shown is a quadratic function displayed on a Cartesian plane with both x- and y-axis ranges from -10 to 10. It is a U-shaped curve opening upwards, which indicates it is a parabola. The vertex of the parabola appears to be at the point $(-2, 1)$, making this the minimum point of the function.

#### Detailed Features:
1. **Axes:**
- The horizontal axis (x-axis) and vertical axis (y-axis) intersect at the origin point (0, 0).
- The x-axis and y-axis are labeled with increments of 1 unit, covering the range from -10 to 10.

2. **Parabola:**
- **Vertex:** The vertex, which is the lowest point of the parabola, is at $(-2, 1)$.
- **Symmetry:** The parabola is symmetric with respect to the vertical line passing through its vertex. This means that if the x-coordinates of points are equidistant from $-2$ on the horizontal axis, their y-coordinates will be the same.
- **Direction:** The parabolic curve opens upwards, implying that the coefficient of the $x^2$ term in its quadratic equation is positive.

3. **Intersection Points with Axes:**
- **Y-axis:** The parabola intersects the y-axis at $ (0, 4) $, indicating that when $ x = 0 $, $ y = 4 $.
- **X-axis:** It intersects the x-axis at two points: approximately $ (-4, 0) $ and $ (0, 0) $. These are the roots or solutions of the quadratic equation when $ y = 0 $.

#### Educational Context:
This graph is an example of a quadratic function, which is fundamental in algebra and pre-calculus. Quadratic functions have the general form $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants. Understanding how to graph these functions and identify their key features, such as the vertex, axis of symmetry, and intercepts, is crucial for students studying quadratic relationships.

#### Key Takeaways:
- The graph of a quadratic function is always a parabola.
- The direction in which the parabola opens (upwards or

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