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

Understanding the Quadratic Function

The quadratic function $f(x) = x^2 + 2x - 3$ is a polynomial equation of degree two, which means it has a parabolic shape. The x-intercept of a quadratic function is the point where the graph of the function crosses the x-axis. In other words, it is the value of x at which the function equals zero.

Finding the X-Intercept

To find the x-intercept of the quadratic function $f(x) = x^2 + 2x - 3$, we need to set the function equal to zero and solve for x. This can be done by factoring the quadratic expression or by using the quadratic formula.

Factoring the Quadratic Expression

We can start by factoring the quadratic expression $x^2 + 2x - 3$. We need to find two numbers whose product is -3 and whose sum is 2. These numbers are 3 and -1, so we can write the quadratic expression as $(x + 3)(x - 1)$.

Setting the Function Equal to Zero

Now that we have factored the quadratic expression, we can set the function equal to zero and solve for x. We have:

$f(x) = (x + 3)(x - 1) = 0$

This equation is true when either $(x + 3) = 0$ or $(x - 1) = 0$.

Solving for x

We can solve for x by setting each factor equal to zero and solving for x. We have:

$x + 3 = 0 \Rightarrow x = -3$

$x - 1 = 0 \Rightarrow x = 1$

Therefore, the x-intercept of the quadratic function $f(x) = x^2 + 2x - 3$ is $x = -3$ and $x = 1$.

Conclusion

In conclusion, the x-intercept of the quadratic function $f(x) = x^2 + 2x - 3$ is $x = -3$ and $x = 1$. This means that the graph of the function crosses the x-axis at the points $(-3, 0)$ and $(1, 0)$.

Answer

The x-intercept(s) is/are $x = -3$ and $x = 1$.

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Part (b) - Finding the Y-Intercept

The y-intercept of a quadratic function is the point where the graph of the function crosses the y-axis. In other words, it is the value of y when x is equal to zero.

Finding the Y-Intercept

To find the y-intercept of the quadratic function $f(x) = x^2 + 2x - 3$, we need to substitute x = 0 into the function and solve for y.

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

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

Conclusion

In conclusion, the y-intercept of the quadratic function $f(x) = x^2 + 2x - 3$ is $y = -3$.

Answer

The y-intercept is $y = -3$.

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Part (c) - Finding the Vertex

The vertex of a quadratic function is the point on the graph of the function that is the lowest or highest point. In other words, it is the point on the graph where the function changes from decreasing to increasing or from increasing to decreasing.

Finding the Vertex

To find the vertex of the quadratic function $f(x) = x^2 + 2x - 3$, we need to use the formula for the x-coordinate of the vertex, which is given by:

$x = -\frac{b}{2a}$

In this case, a = 1 and b = 2, so we have:

$x = -\frac{2}{2(1)} = -1$

Now that we have found the x-coordinate of the vertex, we can substitute this value into the function to find the y-coordinate of the vertex.

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

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

Conclusion

In conclusion, the vertex of the quadratic function $f(x) = x^2 + 2x - 3$ is $(-1, -4)$.

Answer

The vertex is $(-1, -4)$.

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Part (d) - Finding the Axis of Symmetry

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

Finding the Axis of Symmetry

To find the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$, we need to use the formula for the axis of symmetry, which is given by:

$x = -\frac{b}{2a}$

In this case, a = 1 and b = 2, so we have:

$x = -\frac{2}{2(1)} = -1$

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

Conclusion

In conclusion, the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$ is the vertical line $x = -1$.

Answer

The axis of symmetry is $x = -1$.

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Part (e) - Finding the Maximum or Minimum Value

The maximum or minimum value of a quadratic function is the value of the function at the vertex of the function. In other words, it is the highest or lowest point on the graph of the function.

Finding the Maximum or Minimum Value

To find the maximum or minimum value of the quadratic function $f(x) = x^2 + 2x - 3$, we need to substitute the x-coordinate of the vertex into the function and solve for y.

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

Therefore, the maximum or minimum value of the quadratic function $f(x) = x^2 + 2x - 3$ is $y = -4$.

Conclusion

In conclusion, the maximum or minimum value of the quadratic function $f(x) = x^2 + 2x - 3$ is $y = -4$.

Answer

The maximum or minimum value is $y = -4$.

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Part (f) - Finding the Interval of Increase or Decrease

The interval of increase or decrease of a quadratic function is the interval on the x-axis where the function is increasing or decreasing. In other words, it is the interval on the x-axis where the graph of the function is moving upwards or downwards.

Finding the Interval of Increase or Decrease

To find the interval of increase or decrease of the quadratic function $f(x) = x^2 + 2x - 3$, we need to examine the sign of the derivative of the function.

The derivative of the function is given by:

$f'(x) = 2x + 2$

We can examine the sign of the derivative by choosing a test point. Let's choose x = 0.

$f'(0) = 2(0) + 2 = 2$

Since the derivative is positive at x = 0, we know that the function is increasing on the interval $(-\infty, 0)$.

Now, let's choose x = 1.

$f'(1) = 2(1) + 2 = 4$

Since the derivative is positive at x = 1, we know that the function is increasing on the interval $(0, \infty)$.

Therefore, the interval of increase or decrease of the quadratic function $f(x) = x^2 + 2x - 3$ is $(-\infty, 0) \cup (0, \infty)$.

Conclusion

In conclusion, the interval of increase or decrease of the quadratic function $f(x) = x^2 + 2x - 3$ is $(-\infty, 0) \cup (0, \infty)$.

Answer

The interval of increase or decrease is $(-\infty, 0) \cup (0, \infty)$.

Understanding Quadratic Functions

A quadratic function is a polynomial equation of degree two, which means it has a parabolic shape. Quadratic functions are commonly used in mathematics, physics, and engineering to model real-world problems.

Quadratic Function Formula

The general formula for a quadratic function is:

$f(x) = ax^2 + bx + c$

where a, b, and c are constants.

Quadratic Function Examples

Some examples of quadratic functions include:

• $f(x) = x^2 + 2x - 3$
• $f(x) = 2x^2 - 5x + 1$
• $f(x) = x^2 - 4$

Quadratic Function Properties

Quadratic functions have several important properties, including:

• Axis of Symmetry: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex of the function.
• Vertex: The vertex of a quadratic function is the point on the graph of the function that is the lowest or highest point.
• X-Intercepts: The x-intercepts of a quadratic function are the points on the graph of the function where the function crosses the x-axis.
• Y-Intercept: The y-intercept of a quadratic function is the point on the graph of the function where the function crosses the y-axis.

Quadratic Function Applications

Quadratic functions have many real-world applications, including:

• Physics: Quadratic functions are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic functions are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic functions are used to model the behavior of economic systems, such as supply and demand.

Quadratic Function Q&A

Q: What is the x-intercept of the quadratic function $f(x) = x^2 + 2x - 3$?

A: The x-intercept of the quadratic function $f(x) = x^2 + 2x - 3$ is $x = -3$ and $x = 1$.

Q: What is the y-intercept of the quadratic function $f(x) = x^2 + 2x - 3$?

A: The y-intercept of the quadratic function $f(x) = x^2 + 2x - 3$ is $y = -3$.

Q: What is the vertex of the quadratic function $f(x) = x^2 + 2x - 3$?

A: The vertex of the quadratic function $f(x) = x^2 + 2x - 3$ is $(-1, -4)$.

Q: What is the axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$?

A: The axis of symmetry of the quadratic function $f(x) = x^2 + 2x - 3$ is the vertical line $x = -1$.

Q: What is the interval of increase or decrease of the quadratic function $f(x) = x^2 + 2x - 3$?

A: The interval of increase or decrease of the quadratic function $f(x) = x^2 + 2x - 3$ is $(-\infty, 0) \cup (0, \infty)$.

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

A: To find the x-intercept of a quadratic function, you need to set the function equal to zero and solve for x.

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

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

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

A: To find the vertex of a quadratic function, you need to use the formula for the x-coordinate of the vertex, which is given by:

$x = -\frac{b}{2a}$

Then, you need to substitute this value into the function to find the y-coordinate of the vertex.

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

A: To find the axis of symmetry of a quadratic function, you need to use the formula for the axis of symmetry, which is given by:

$x = -\frac{b}{2a}$

Q: How do I find the interval of increase or decrease of a quadratic function?

A: To find the interval of increase or decrease of a quadratic function, you need to examine the sign of the derivative of the function.

Conclusion

In conclusion, quadratic functions are an important topic in mathematics, and understanding their properties and applications is crucial for solving real-world problems. By following the steps outlined in this article, you can find the x-intercept, y-intercept, vertex, axis of symmetry, and interval of increase or decrease of a quadratic function.