Is { G $}$ The Graph Of { F(x) = 4x^2 - 12x + 14 $}$?

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Introduction

In mathematics, graphing functions is a crucial aspect of understanding their behavior and properties. Given a function, we can represent it graphically, which helps in visualizing its characteristics, such as its maximum and minimum values, intervals of increase and decrease, and points of inflection. In this article, we will explore whether a given graph, { g $}$, represents the function { f(x) = 4x^2 - 12x + 14 $}$. To determine this, we need to analyze the properties of the function and compare them with the characteristics of the graph.

Understanding the Function

The given function is a quadratic function in the form of { f(x) = ax^2 + bx + c $}$, where { a = 4 $}$, { b = -12 $}$, and { c = 14 $}$. To understand the behavior of this function, we need to examine its vertex, axis of symmetry, and the direction of its opening.

Vertex and Axis of Symmetry

The vertex of a quadratic function is the point where the function changes direction, and it is given by the formula { x = -\fracb}{2a} $}$. In this case, { x = -\frac{-12}{2(4)} = \frac{12}{8} = \frac{3}{2} $}$. To find the y-coordinate of the vertex, we substitute this value of x into the function { f(\frac{3{2}) = 4(\frac{3}{2})^2 - 12(\frac{3}{2}) + 14 $}$. Simplifying this expression, we get { f(\frac{3}{2}) = 4(\frac{9}{4}) - 18 + 14 = 9 - 18 + 14 = 5 $}$. Therefore, the vertex of the function is at the point { (\frac{3}{2}, 5) $}$.

The axis of symmetry is a vertical line that passes through the vertex, and its equation is given by { x = -\frac{b}{2a} $}$. In this case, the axis of symmetry is { x = \frac{3}{2} $}$.

Direction of Opening

The direction of opening of a quadratic function is determined by the sign of the coefficient of the squared term, which is { a $}$. In this case, { a = 4 $}$, which is positive. Therefore, the function opens upward.

Analyzing the Graph

To determine whether the graph { g $}$ represents the function { f(x) = 4x^2 - 12x + 14 $}$, we need to analyze its properties and compare them with the characteristics of the function.

Vertex and Axis of Symmetry

The vertex of the graph { g $}$ is at the point { (\frac{3}{2}, 5) $}$, which matches the vertex of the function { f(x) = 4x^2 - 12x + 14 $}$. The axis of symmetry of the graph is also { x = \frac{3}{2} $}$, which is the same as the axis of symmetry of the function.

Direction of Opening

The graph { g $}$ opens upward, which is the same direction as the function { f(x) = 4x^2 - 12x + 14 $}$.

Conclusion

Based on the analysis of the properties of the function and the graph, we can conclude that the graph { g $}$ represents the function { f(x) = 4x^2 - 12x + 14 $}$. The vertex, axis of symmetry, and direction of opening of the graph match the characteristics of the function, which confirms that the graph is indeed the representation of the function.

Final Thoughts

Graphing functions is an essential aspect of mathematics, and understanding the properties of a function is crucial in determining its behavior and characteristics. By analyzing the vertex, axis of symmetry, and direction of opening of a function, we can determine whether a given graph represents the function. In this article, we have demonstrated how to analyze the properties of a function and compare them with the characteristics of a graph to determine whether the graph represents the function.

References

Additional Resources

Introduction

In our previous article, we explored whether a given graph, { g $}$, represents the function { f(x) = 4x^2 - 12x + 14 $}$. We analyzed the properties of the function and compared them with the characteristics of the graph to determine whether the graph represents the function. In this article, we will answer some frequently asked questions related to graphing functions and quadratic functions.

Q&A

Q1: What is the vertex of the function { f(x) = 4x^2 - 12x + 14 $}$?

A1: The vertex of the function { f(x) = 4x^2 - 12x + 14 $}$ is at the point { (\frac3}{2}, 5) $}$. To find the vertex, we use the formula { x = -\frac{b}{2a} $}$, where { a = 4 $}$ and { b = -12 $}$. Substituting these values, we get { x = -\frac{-12}{2(4)} = \frac{12}{8} = \frac{3}{2} $}$. To find the y-coordinate of the vertex, we substitute this value of x into the function { f(\frac{3{2}) = 4(\frac{3}{2})^2 - 12(\frac{3}{2}) + 14 $}$. Simplifying this expression, we get { f(\frac{3}{2}) = 4(\frac{9}{4}) - 18 + 14 = 9 - 18 + 14 = 5 $}$.

Q2: What is the axis of symmetry of the function { f(x) = 4x^2 - 12x + 14 $}$?

A2: The axis of symmetry of the function { f(x) = 4x^2 - 12x + 14 $}$ is the vertical line { x = \frac{3}{2} $}$. To find the axis of symmetry, we use the formula { x = -\frac{b}{2a} $}$, where { a = 4 $}$ and { b = -12 $}$. Substituting these values, we get { x = -\frac{-12}{2(4)} = \frac{12}{8} = \frac{3}{2} $}$.

Q3: Does the graph { g $}$ open upward or downward?

A3: The graph { g $}$ opens upward. To determine the direction of opening, we examine the sign of the coefficient of the squared term, which is { a = 4 $}$. Since { a $}$ is positive, the graph opens upward.

Q4: How do we find the x-intercepts of the function { f(x) = 4x^2 - 12x + 14 $}$?

A4: To find the x-intercepts of the function { f(x) = 4x^2 - 12x + 14 $}$, we set the function equal to zero and solve for x: { 4x^2 - 12x + 14 = 0 $}$. We can use the quadratic formula to solve for x: { x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $}$, where { a = 4 $}$, { b = -12 $}$, and { c = 14 $}$. Substituting these values, we get { x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(14)}}{2(4)} $}$. Simplifying this expression, we get { x = \frac{12 \pm \sqrt{144 - 224}}{8} $}$. Further simplifying, we get { x = \frac{12 \pm \sqrt{-80}}{8} $}$. Since the square root of a negative number is not a real number, the function has no real x-intercepts.

Q5: How do we find the y-intercept of the function { f(x) = 4x^2 - 12x + 14 $}$?

A5: To find the y-intercept of the function { f(x) = 4x^2 - 12x + 14 $}$, we substitute x = 0 into the function: { f(0) = 4(0)^2 - 12(0) + 14 $}$. Simplifying this expression, we get { f(0) = 14 $}$. Therefore, the y-intercept of the function is 14.

Conclusion

In this article, we have answered some frequently asked questions related to graphing functions and quadratic functions. We have explored the properties of the function { f(x) = 4x^2 - 12x + 14 $}$ and compared them with the characteristics of the graph { g $}$. We have also provided step-by-step solutions to some common problems related to graphing functions and quadratic functions.

Final Thoughts

Graphing functions is an essential aspect of mathematics, and understanding the properties of a function is crucial in determining its behavior and characteristics. By analyzing the vertex, axis of symmetry, and direction of opening of a function, we can determine whether a given graph represents the function. In this article, we have demonstrated how to analyze the properties of a function and compare them with the characteristics of a graph to determine whether the graph represents the function.

References

Additional Resources