For The Quadratic Function F ( X ) = X 2 + 2 X − 3 F(x) = X^2 + 2x - 3 F ( X ) = X 2 + 2 X − 3 , Answer Parts (a) Through (f).(a) What Is The X-intercept? Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. The X-intercept(s) Is/are (1, 0),

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 and answer parts (a) through (f) of a given problem.

Part (a): Finding the x-Intercept

The x-intercept of a quadratic function is the point where the graph of the function intersects the x-axis. In other words, it is the value of x when y = 0. To find the x-intercept, we need to set the function equal to zero and solve for x.

Step 1: Set the function equal to zero

f(x)=x2+2x−3=0f(x) = x^2 + 2x - 3 = 0

Step 2: Factor the quadratic expression

x2+2x−3=(x+3)(x−1)=0x^2 + 2x - 3 = (x + 3)(x - 1) = 0

Step 3: Solve for x

(x+3)=0(x + 3) = 0 or (x−1)=0(x - 1) = 0

x+3=0⇒x=−3x + 3 = 0 \Rightarrow x = -3

x−1=0⇒x=1x - 1 = 0 \Rightarrow x = 1

Therefore, the x-intercept(s) of the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is/are (1, 0).

Part (b): Finding the y-Intercept

The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. In other words, it is the value of y when x = 0. To find the y-intercept, we need to substitute x = 0 into the function.

Step 1: Substitute x = 0 into the function

f(0)=(0)2+2(0)−3=−3f(0) = (0)^2 + 2(0) - 3 = -3

Therefore, the y-intercept of the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is (-3, 0).

Part (c): Finding the Vertex

The vertex of a quadratic function is the point where the graph of the function changes direction. In other words, it is the minimum or maximum point of the graph. To find the vertex, we need to use the formula:

x=−b2ax = -\frac{b}{2a}

where a and b are the coefficients of the quadratic expression.

Step 1: Identify the coefficients a and b

a=1a = 1 and b=2b = 2

Step 2: Substitute the values of a and b into the formula

x=−22(1)=−1x = -\frac{2}{2(1)} = -1

Step 3: Find the value of y by substituting x = -1 into the function

f(−1)=(−1)2+2(−1)−3=−4f(-1) = (-1)^2 + 2(-1) - 3 = -4

Therefore, the vertex of the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is (-1, -4).

Part (d): Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. In other words, it is the line that divides the graph of the function into two symmetrical parts. To find the axis of symmetry, we need to use the formula:

x=−b2ax = -\frac{b}{2a}

where a and b are the coefficients of the quadratic expression.

Step 1: Identify the coefficients a and b

a=1a = 1 and b=2b = 2

Step 2: Substitute the values of a and b into the formula

x=−22(1)=−1x = -\frac{2}{2(1)} = -1

Therefore, the axis of symmetry of the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is x = -1.

Part (e): Finding the Domain

The domain of a quadratic function is the set of all possible input values (x-values) that the function can accept. In other words, it is the set of all possible x-values that make the function defined. Since the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is a polynomial function, its domain is all real numbers.

Part (f): Finding the Range

The range of a quadratic function is the set of all possible output values (y-values) that the function can produce. In other words, it is the set of all possible y-values that make the function defined. Since the quadratic function f(x)=x2+2x−3f(x) = x^2 + 2x - 3 is a polynomial function, its range is all real numbers.

Conclusion

Frequently Asked Questions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has the general form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the x-intercept of a quadratic function?

A: The x-intercept of a quadratic function is the point where the graph of the function intersects the x-axis. In other words, it is the value of x when y = 0.

Q: How do I find the x-intercept of a quadratic function?

A: To find the x-intercept, you need to set the function equal to zero and solve for x. You can use factoring, the quadratic formula, or other methods to solve for x.

Q: What is the y-intercept of a quadratic function?

A: The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. In other words, it is the value of y when x = 0.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept, you need to substitute x = 0 into the function and solve for y.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the graph of the function changes direction. In other words, it is the minimum or maximum point of the graph.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex, you need to use the formula x=−b2ax = -\frac{b}{2a}, where a and b are the coefficients of the quadratic expression.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. In other words, it is the line that divides the graph of the function into two symmetrical parts.

Q: How do I find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry, you need to use the formula x=−b2ax = -\frac{b}{2a}, where a and b are the coefficients of the quadratic expression.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values (x-values) that the function can accept. In other words, it is the set of all possible x-values that make the function defined.

Q: How do I find the domain of a quadratic function?

A: Since quadratic functions are polynomial functions, their domain is all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values (y-values) that the function can produce. In other words, it is the set of all possible y-values that make the function defined.

Q: How do I find the range of a quadratic function?

A: Since quadratic functions are polynomial functions, their range is all real numbers.

Q: Can I use a calculator to solve quadratic equations?

A: Yes, you can use a calculator to solve quadratic equations. Most calculators have a built-in quadratic formula function that you can use to solve for x.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot points on the graph.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including physics, engineering, economics, and more. Some examples include:

  • Projectile motion
  • Optimization problems
  • Quadratic equations in finance
  • Quadratic equations in computer science

Q: Can I use quadratic functions to model real-world phenomena?

A: Yes, you can use quadratic functions to model real-world phenomena. Quadratic functions can be used to model situations where there is a maximum or minimum value, such as the height of a projectile or the cost of a product.

Q: How do I determine if a quadratic function is a good model for a real-world phenomenon?

A: To determine if a quadratic function is a good model for a real-world phenomenon, you need to check if the function accurately represents the data and if it has a minimum or maximum value that makes sense in the context of the problem.