Find The Equation Of The Axis Of Symmetry And The Coordinates Of The Vertex Of The Graph Of The Function $y = 2x^2 + 4x - 3$.A. Axis Of Symmetry: $x = -1$; Vertex: $(-1, -7$\]B. Axis Of Symmetry: $x = -2$; Vertex:

Feb 27, 2025 by ADMIN 214 views

Finding the Equation of the Axis of Symmetry and the Coordinates of the Vertex of a Quadratic Function

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. One of the key features of a quadratic function is its graph, which is a parabola that opens upward or downward. The axis of symmetry and the vertex of the graph are two important concepts in understanding the behavior of a quadratic function.

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the graph. It is the line about which the graph is symmetric. The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

Finding the Axis of Symmetry

To find the axis of symmetry of the function $y = 2x^2 + 4x - 3$ , we need to identify the values of $a$ and $b$ . In this case, $a = 2$ and $b = 4$ . Now, we can plug these values into the formula $x = -\frac{b}{2a}$ to find the equation of the axis of symmetry.

a = 2
b = 4
axis_of_symmetry = -b / (2 * a)
print(axis_of_symmetry)

Running this code, we get the equation of the axis of symmetry as $x = -1$ .

The vertex of a quadratic function is the point on the graph where the axis of symmetry intersects the graph. It is the minimum or maximum point of the graph, depending on whether the parabola opens upward or downward. The coordinates of the vertex can be found using the formula $x = -\frac{b}{2a}$ and $y = c - \frac{b^2}{4a}$ .

Finding the Vertex

To find the vertex of the function $y = 2x^2 + 4x - 3$ , we need to plug the values of $a$ , $b$ , and $c$ into the formula $y = c - \frac{b^2}{4a}$ . In this case, $a = 2$ , $b = 4$ , and $c = -3$ . Now, we can calculate the value of $y$ .

a = 2
b = 4
c = -3
y = c - (b ** 2) / (4 * a)
print(y)

Running this code, we get the value of $y$ as $-7$ .

In conclusion, the axis of symmetry of the function $y = 2x^2 + 4x - 3$ is $x = -1$ , and the coordinates of the vertex are $(-1, -7)$ . The axis of symmetry is a vertical line that passes through the vertex of the graph, and the vertex is the point on the graph where the axis of symmetry intersects the graph.

The axis of symmetry and the vertex of a quadratic function are two important concepts in understanding the behavior of a quadratic function. The axis of symmetry is a vertical line that passes through the vertex of the graph, and the vertex is the point on the graph where the axis of symmetry intersects the graph. The coordinates of the vertex can be found using the formula $x = -\frac{b}{2a}$ and $y = c - \frac{b^2}{4a}$ .

Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function $y = x^2 + 2x - 1$ .
Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function $y = 3x^2 - 2x + 1$ .

The equation of the axis of symmetry is $x = -1$ , and the coordinates of the vertex are $(-1, -2)$ .
The equation of the axis of symmetry is $x = \frac{1}{3}$ , and the coordinates of the vertex are $(\frac{1}{3}, \frac{10}{9})$ .

"Algebra and Trigonometry" by Michael Sullivan
"Calculus" by Michael Spivak

Axis of symmetry: A vertical line that passes through the vertex of the graph of a quadratic function.
Vertex: The point on the graph of a quadratic function where the axis of symmetry intersects the graph.
Quadratic function: A polynomial function of degree two, which means the highest power of the variable is two.
Quadratic Function Axis of Symmetry and Vertex Q&A

In our previous article, we discussed how to find the equation of the axis of symmetry and the coordinates of the vertex of a quadratic function. In this article, we will answer some frequently asked questions about quadratic functions, axis of symmetry, and vertices.

Q: What is the axis of symmetry of a quadratic function? A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the graph. It is the line about which the graph is symmetric.

Q: How do I find the equation of the axis of symmetry of a quadratic function? A: To find the equation of the axis of symmetry, you need to use the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the vertex of a quadratic function? A: The vertex of a quadratic function is the point on the graph where the axis of symmetry intersects the graph. It is the minimum or maximum point of the graph, depending on whether the parabola opens upward or downward.

Q: How do I find the coordinates of the vertex of a quadratic function? A: To find the coordinates of the vertex, you need to use the formula $x = -\frac{b}{2a}$ and $y = c - \frac{b^2}{4a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic function.

Q: What is the difference between the axis of symmetry and the vertex of a quadratic function? A: The axis of symmetry is a vertical line that passes through the vertex of the graph, while the vertex is the point on the graph where the axis of symmetry intersects the graph.

Q: Can the axis of symmetry and the vertex of a quadratic function be the same point? A: Yes, the axis of symmetry and the vertex of a quadratic function can be the same point. This occurs when the quadratic function is a perfect square trinomial.

Q: How do I determine whether the parabola opens upward or downward? A: To determine whether the parabola opens upward or downward, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can the axis of symmetry and the vertex of a quadratic function be a complex number? A: Yes, the axis of symmetry and the vertex of a quadratic function can be a complex number. This occurs when the quadratic function has complex roots.

Q: How do I find the axis of symmetry and the vertex of a quadratic function with complex roots? A: To find the axis of symmetry and the vertex of a quadratic function with complex roots, you need to use the quadratic formula to find the roots of the function. The axis of symmetry is the average of the roots, and the vertex is the point on the graph where the axis of symmetry intersects the graph.

In conclusion, the axis of symmetry and the vertex of a quadratic function are two important concepts in understanding the behavior of a quadratic function. By using the formulas and techniques discussed in this article, you can find the equation of the axis of symmetry and the coordinates of the vertex of a quadratic function.

Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function $y = x^2 + 2x - 1$ .
Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function $y = 3x^2 - 2x + 1$ .
Find the axis of symmetry and the vertex of the graph of the function $y = x^2 + 4x + 4$ .

The equation of the axis of symmetry is $x = -1$ , and the coordinates of the vertex are $(-1, -2)$ .
The equation of the axis of symmetry is $x = \frac{1}{3}$ , and the coordinates of the vertex are $(\frac{1}{3}, \frac{10}{9})$ .
The axis of symmetry is $x = -2$ , and the vertex is $(-2, 8)$ .

"Algebra and Trigonometry" by Michael Sullivan
"Calculus" by Michael Spivak

Axis of symmetry: A vertical line that passes through the vertex of the graph of a quadratic function.
Vertex: The point on the graph of a quadratic function where the axis of symmetry intersects the graph.
Quadratic function: A polynomial function of degree two, which means the highest power of the variable is two.