Unlocking H(x) Finding Zeros And Evaluating At Zero
In the realm of mathematics, quadratic functions hold a special significance, weaving their way through various applications, from projectile motion to optimization problems. To truly grasp the essence of these functions, it's essential to delve into their properties and uncover their hidden characteristics. This exploration delves into the quadratic function h(x) = x² + 4x + 3, aiming to decipher its behavior and extract meaningful insights. We will determine the values of x for which h(x) = 0, which are the roots or zeros of the function, and evaluate h(0), which reveals the function's y-intercept. By dissecting this function, we'll gain a deeper appreciation for the elegance and power of quadratic equations.
Part (a): Finding the Zeros of h(x) - Where the Function Intersects the x-axis
The first part of our exploration focuses on pinpointing the values of x that make h(x) = 0. These values, often referred to as the zeros or roots of the function, hold paramount importance as they represent the points where the parabola intersects the x-axis. Unveiling these zeros allows us to understand the function's behavior around the x-axis and provides crucial information for graphing and analyzing the quadratic equation. To embark on this quest, we'll employ the time-tested technique of factoring, which allows us to rewrite the quadratic expression as a product of two linear factors. This transformation simplifies the equation and allows us to isolate the values of x that satisfy the condition h(x) = 0.
When we set h(x) = 0, we're essentially seeking the x-values that make the equation x² + 4x + 3 = 0 true. To solve this, we need to factor the quadratic expression x² + 4x + 3. Factoring involves finding two numbers that add up to the coefficient of the x term (which is 4 in this case) and multiply to the constant term (which is 3). After careful consideration, we identify that the numbers 1 and 3 fit the bill perfectly, as 1 + 3 = 4 and 1 * 3 = 3. Now, we can rewrite the quadratic expression in its factored form: (x + 1)(x + 3) = 0.
The beauty of factoring lies in its ability to transform a complex equation into a simpler one. When the product of two factors equals zero, it implies that at least one of the factors must be zero. Therefore, to satisfy the equation (x + 1)(x + 3) = 0, either (x + 1) = 0 or (x + 3) = 0. Solving these two linear equations separately, we find the solutions: x = -1 and x = -3. These solutions are the zeros of the function h(x), representing the x-coordinates where the parabola intersects the x-axis. Geometrically, these points mark the locations where the parabola crosses the horizontal axis, providing crucial information about the function's graph and behavior. Therefore, we can confidently assert that the values of x for which h(x) = 0 are -1 and -3.
Part (b): Evaluating h(0) - Unveiling the Function's Y-Intercept
In the second part of our investigation, we shift our focus to evaluating h(0). This evaluation provides us with the value of the function when x is set to zero. This seemingly simple calculation holds significant geometric meaning, as h(0) represents the y-intercept of the quadratic function's graph. The y-intercept is the point where the parabola intersects the y-axis, offering valuable information about the function's vertical position and behavior. By determining h(0), we gain a crucial anchor point for sketching the graph of the function and understanding its overall characteristics. To find h(0), we simply substitute x = 0 into the function's equation and perform the necessary calculations.
To find h(0), we substitute x = 0 into the function h(x) = x² + 4x + 3, resulting in h(0) = (0)² + 4(0) + 3. Simplifying this expression, we find that h(0) = 0 + 0 + 3 = 3. This result signifies that when x is zero, the function's value is 3. Geometrically, this corresponds to the point (0, 3) on the Cartesian plane, which is precisely where the parabola intersects the y-axis. The y-intercept, therefore, provides a crucial reference point for visualizing the graph of the quadratic function. It tells us the height at which the parabola crosses the vertical axis, adding to our understanding of its overall shape and position. In essence, h(0) = 3 reveals the function's y-intercept, marking the spot where the parabola gracefully intersects the y-axis.
Summary: A Comprehensive Understanding of h(x) = x² + 4x + 3
In this exploration, we've dissected the quadratic function h(x) = x² + 4x + 3 to uncover its key characteristics. We've successfully determined the values of x for which h(x) = 0, finding the zeros of the function to be -1 and -3. These zeros represent the points where the parabola intersects the x-axis, providing crucial insights into the function's behavior around the horizontal axis. Furthermore, we've evaluated h(0), discovering the function's y-intercept to be 3. This value signifies the point (0, 3) where the parabola intersects the y-axis, offering a vital reference point for visualizing the function's graph. By combining these findings, we've gained a comprehensive understanding of the quadratic function h(x) = x² + 4x + 3, paving the way for further analysis and applications.
This exploration demonstrates the power of mathematical tools in unraveling the mysteries of functions. By finding the zeros and y-intercept, we've gained a solid foundation for understanding the behavior of this quadratic function. These concepts are fundamental in various mathematical contexts and serve as stepping stones for more advanced explorations in algebra and calculus.
