Graphing Quadratic Functions Plotting Vertex And Intercepts For F(x) = 4x^2 - 8x + 3

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In the realm of mathematics, quadratic functions hold a significant place, serving as fundamental tools for modeling various real-world phenomena. From projectile motion to the curvature of suspension bridges, quadratic functions provide a powerful means of understanding and predicting parabolic trajectories. In this comprehensive guide, we will delve into the intricacies of graphing a quadratic function by meticulously plotting its vertex and intercepts. Our focus will be on the specific function $f(x) = 4x^2 - 8x + 3$, providing a step-by-step approach that empowers you to visualize and interpret quadratic functions with confidence.

Understanding Quadratic Functions

Before we embark on the graphing process, it's crucial to grasp the fundamental characteristics of quadratic functions. A quadratic function is defined as a polynomial function of degree two, generally expressed in the form $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola's orientation, whether it opens upwards or downwards, is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards.

Key Features of a Parabola

• Vertex: The vertex represents the minimum or maximum point of the parabola. For a parabola opening upwards, the vertex is the minimum point, while for a parabola opening downwards, the vertex is the maximum point. The x-coordinate of the vertex is given by the formula $x = -b / 2a$, and the y-coordinate can be found by substituting this x-value back into the function.
• Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is given by $x = -b / 2a$, which is the same as the x-coordinate of the vertex.
• Intercepts: Intercepts are the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept). The x-intercepts, also known as roots or zeros, are the solutions to the quadratic equation $f(x) = 0$. They can be found by factoring the quadratic expression, using the quadratic formula, or completing the square. The y-intercept is the point where the parabola intersects the y-axis, which occurs when $x = 0$. It can be found by substituting $x = 0$ into the function.

Step-by-Step Guide to Graphing $f(x) = 4x^2 - 8x + 3$

Now that we have a solid understanding of quadratic functions and their key features, let's apply this knowledge to graph the specific function $f(x) = 4x^2 - 8x + 3$. We will follow a systematic approach, meticulously plotting the vertex and intercepts to accurately represent the parabola.

Step 1: Determine the Orientation of the Parabola

The first step is to identify the coefficient 'a' in the quadratic function. In our case, $f(x) = 4x^2 - 8x + 3$, the coefficient 'a' is 4. Since 'a' is positive, the parabola opens upwards, indicating that the vertex will be the minimum point.

Step 2: Find the Vertex

To find the vertex, we will use the formula $x = -b / 2a$ to determine the x-coordinate and then substitute this value back into the function to find the y-coordinate.

In our function, $f(x) = 4x^2 - 8x + 3$, 'b' is -8 and 'a' is 4. Plugging these values into the formula, we get:

$x = -(-8) / (2 * 4) = 8 / 8 = 1$

Now, substitute $x = 1$ back into the function to find the y-coordinate:

$f(1) = 4(1)^2 - 8(1) + 3 = 4 - 8 + 3 = -1$

Therefore, the vertex of the parabola is at the point (1, -1).

Step 3: Find the Intercepts

Next, we will determine the intercepts, which are the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept).

Finding the x-intercepts:

To find the x-intercepts, we need to solve the quadratic equation $f(x) = 0$:

$4x^2 - 8x + 3 = 0$

We can solve this equation by factoring, using the quadratic formula, or completing the square. In this case, let's try factoring:

$(2x - 1)(2x - 3) = 0$

Setting each factor equal to zero, we get:

$2x - 1 = 0$ or $2x - 3 = 0$

Solving for x, we find:

$x = 1/2$ or $x = 3/2$

Therefore, the x-intercepts are (1/2, 0) and (3/2, 0).

Finding the y-intercept:

To find the y-intercept, we need to substitute $x = 0$ into the function:

$f(0) = 4(0)^2 - 8(0) + 3 = 3$

Therefore, the y-intercept is (0, 3).

Step 4: Plot the Vertex and Intercepts

Now that we have found the vertex (1, -1), the x-intercepts (1/2, 0) and (3/2, 0), and the y-intercept (0, 3), we can plot these points on a coordinate plane.

Step 5: Sketch the Parabola

Finally, we can sketch the parabola by drawing a smooth U-shaped curve that passes through the plotted points. Remember that the parabola is symmetrical about the axis of symmetry, which is the vertical line passing through the vertex. In this case, the axis of symmetry is the line $x = 1$.

Visualizing the Graph

By plotting the vertex and intercepts, we have successfully visualized the graph of the quadratic function $f(x) = 4x^2 - 8x + 3$. The parabola opens upwards, with its vertex at (1, -1), x-intercepts at (1/2, 0) and (3/2, 0), and y-intercept at (0, 3). This graphical representation provides a clear understanding of the function's behavior and its relationship to the coordinate plane.

Alternative Methods for Graphing Quadratic Functions

While plotting the vertex and intercepts is a fundamental method for graphing quadratic functions, there are alternative approaches that can provide additional insights and perspectives.

1. Using the Standard Form of a Quadratic Equation

The standard form of a quadratic equation is $f(x) = a(x - h)^2 + k$, where (h, k) represents the vertex of the parabola. By converting the given quadratic function into standard form, we can directly identify the vertex and readily graph the parabola. To convert $f(x) = 4x^2 - 8x + 3$ into standard form, we can complete the square:

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

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

$f(x) = 4(x - 1)^2 - 1$

From this standard form, we can see that the vertex is indeed (1, -1), which confirms our previous calculations.

2. Using Transformations

Another approach to graphing quadratic functions involves understanding transformations of the basic parabola $f(x) = x^2$. By recognizing the transformations applied to the basic parabola, we can efficiently sketch the graph of the given function. For example, the function $f(x) = 4x^2 - 8x + 3$ can be seen as a transformation of the basic parabola $f(x) = x^2$:

• Vertical Stretch: The coefficient 'a' = 4 stretches the parabola vertically by a factor of 4.
• Horizontal Shift: The term $(x - 1)^2$ shifts the parabola 1 unit to the right.
• Vertical Shift: The constant term -1 shifts the parabola 1 unit downwards.

By understanding these transformations, we can accurately sketch the graph of the function without explicitly plotting points.

Conclusion

Graphing quadratic functions is a fundamental skill in mathematics, providing a visual representation of their behavior and properties. By meticulously plotting the vertex and intercepts, we can accurately sketch the parabola and gain a deeper understanding of the function's characteristics. In this guide, we have provided a step-by-step approach to graphing the quadratic function $f(x) = 4x^2 - 8x + 3$, empowering you to confidently visualize and interpret quadratic functions. Additionally, we have explored alternative methods for graphing quadratic functions, such as using the standard form and transformations, further enhancing your understanding and problem-solving capabilities. With practice and a solid grasp of the concepts discussed, you will be well-equipped to tackle any quadratic graphing challenge.