A Quadratic Function F Is Defined By F(x)=(x-7)(x+3). Identify The X-intercepts Of The Graph Of F

A Quadratic Function F Is Defined By F(x)=(x-7)(x+3). Identify The X-intercepts Of The Graph Of F

Understanding quadratic functions is fundamental in algebra and helps us analyze the behavior of parabolas, their roots, and their intercepts. In this article, we will explore the quadratic function defined by F(x) = (x - 7)(x + 3), focusing primarily on identifying the x-intercepts of its graph. We will break down the process step-by-step, covering key concepts such as factored forms, x-intercepts, and how to find them.

What Is a Quadratic Function?

A quadratic function is a polynomial function of degree 2, typically expressed in the form:

• Standard form: ax² + bx + c
• Factored form: a(x - r₁)(x - r₂)
• Vertex form: a(x - h)² + k

In the context of our specific function, F(x) = (x - 7)(x + 3), the function is given in factored form. This form is particularly useful for quickly identifying the roots or x-intercepts of the quadratic function.

Understanding the Given Function: F(x) = (x - 7)(x + 3)

The function F(x) is expressed as the product of two binomials. Each binomial contains a linear expression involving x. To analyze this function, especially to find its x-intercepts, it is essential to understand what the roots of the function are.

Factored Form and Its Significance

The factored form of a quadratic function directly reveals its roots. Roots are the values of x for which the function evaluates to zero, i.e., F(x) = 0.

Given F(x) = (x - 7)(x + 3), for F(x) to be zero:

(x - 7)(x + 3) = 0

Applying the Zero Product Property (which states that if a product of two factors equals zero, then at least one of the factors must be zero), we get:

• x - 7 = 0


• x + 3 = 0

Finding the X-Intercepts of the Graph of F

The x-intercepts of a graph are the points where the graph crosses or touches the x-axis. These points occur where the output value, F(x), is zero. Therefore, finding the x-intercepts involves solving F(x) = 0.

Step-by-Step Process to Find the X-Intercepts

1. Set the function equal to zero:

1. Apply the Zero Product Property:

1. Solve each equation:

• x - 7 = 0 → x = 7
• x + 3 = 0 → x = -3

1. Identify the x-intercepts:

Therefore, the graph of the quadratic function crosses the x-axis at the points (7, 0) and (-3, 0).

Interpreting the X-Intercepts in Context

The roots of the quadratic function provide vital information about the graph's behavior and the solutions to the related equations.

• At x = 7, the graph intersects the x-axis at the point (7, 0). This indicates that when x is 7, F(x) equals zero.


• At x = -3, the graph intersects the x-axis at the point (-3, 0). This indicates that when x is -3, F(x) equals zero.

Additional Insights: Vertex and Axis of Symmetry

While our primary focus is on x-intercepts, understanding other features of the parabola enhances comprehension.

Finding the Vertex

The vertex of a parabola in factored form can be found by calculating the midpoint between the roots (x-intercepts).

• Midpoint formula: (x₁ + x₂) / 2

(7 + (-3)) / 2 = (4) / 2 = 2

The x-coordinate of the vertex is at x = 2.

To find the y-coordinate of the vertex, substitute x = 2 into the original function:

F(2) = (2 - 7)(2 + 3) = (-5)(5) = -25

Thus, the vertex of the parabola is at (2, -25).

Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex:

x = 2

This line divides the parabola into two symmetric halves.

Converting the Function to Standard Form

While the factored form is excellent for identifying roots, converting to standard form provides additional insights such as the y-intercept and the parabola's shape.

Starting with:

F(x) = (x - 7)(x + 3)

Apply the distributive property (FOIL method):

F(x) = x x + x 3 - 7 x - 7 3

F(x) = x² + 3x - 7x - 21

Combine like terms:

F(x) = x² - 4x - 21

Standard form of the quadratic: F(x) = x² - 4x - 21

From this form, the y-intercept occurs at x = 0:

F(0) = 0 - 0 - 21 = -21

So, the graph crosses the y-axis at (0, -21).

Graphing the Quadratic Function

Using the information obtained, one can sketch the graph of F(x):

• The x-intercepts at (-3, 0) and (7, 0)


• The vertex at (2, -25)


• The y-intercept at (0, -21)


• The axis of symmetry at x = 2

Applications of Finding X-Intercepts

Identifying x-intercepts is a fundamental skill with numerous applications across various fields:

• Physics: Determining points where an object reaches a certain position or velocity.


• Economics: Finding break-even points in profit or loss analysis.


• Engineering: Analyzing systems where certain variables reach specific thresholds.


• Biology: Modeling population dynamics where populations reach zero.

Summary and Key Takeaways

• The quadratic function F(x) = (x - 7)(x + 3) is in factored form, making the identification of roots straightforward.


• The x-intercepts are at x = 7 and x = -3, corresponding to the points where the graph crosses the x-axis.


• Converting to standard form aids in understanding additional features like the vertex and y-intercept.


• Recognizing the symmetry and shape of the parabola helps in graphing and interpreting the function.


• The process of finding x-intercepts involves setting the function equal to zero and solving for x, leveraging the Zero Product Property.

Conclusion

The quadratic function F(x) = (x - 7)(x + 3) exemplifies how factored form simplifies the process of identifying x-intercepts. By setting the function equal to zero and applying fundamental algebraic principles, we find the roots at x = 7 and x = -3. These intercepts are critical in sketching the graph and understanding the behavior of the quadratic function. Whether used in academic settings or real-world applications, mastering the process of finding x-intercepts enhances problem-solving skills and deepens comprehension of quadratic functions.

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Meta Description: Learn how to identify the x-intercepts of the quadratic function F(x) = (x - 7)(x + 3). Step-by-step guide to solving for roots, understanding the graph, and interpreting key features of the parabola.