Question

ANALYZING EQUATIONS The graph of which function has the same axis of symmetry as the graph of \(y=x^2+2 x+2 ?\) (A) \(y=2 x^2+2 x+2\) (B) \(y=-3 x^2-6 x+2\) (C) \(y=x^2-2 x+2\) (D) \(y=-5 x^2+10 x+2\)

Short answer

The function that has the same axis of symmetry as the given function is option (B) \(y = -3x^2 -6x +2\).

Step-by-step solution

Find Axis of Symmetry of the Given Function

Start by finding the axis of symmetry for the given quadratic equation \(y = x^2 + 2x + 2 \) using the formula \(x = -b/(2a)\). Here, \(a = 1\) and \(b = 2\). So, the axis of symmetry of the given function is \(x = -2/(2*1) = -1\).

Find Axis of Symmetry for Each Option

Now, calculate the axis of symmetry for each of the options. For option (A) the axis of symmetry is \(x = -2/(2*2) = -0.5\), for option (B) \(x = - (-6) /(2*-3)= -1\), for option (C) \(x = -(-2)/ (2*1)= 1\), and for option (D) \(x = -10 / (2*-5) = 1\).

Match Axis of Symmetry

The function that has the same axis of symmetry as the given function is the one from option (B), which is \(y = -3x^2 -6x +2\), since both have the axis of symmetry at \(x = -1\).

Key concepts

Quadratic Functions

Quadratic functions are an essential part of algebra and are commonly written in the form \(y = ax^2 + bx + c\). These equations are characterized by their parabolic graphs, which can open upwards or downwards depending on the sign of the coefficient \(a\).
When \(a > 0\), the parabola opens upwards, creating a U-shaped graph. When \(a < 0\), it opens downwards, forming an upside-down U shape. The general shape and direction of the parabola provide a visualized framework to study its properties.
Key features of quadratic functions include their vertex, axis of symmetry, and roots (or x-intercepts). Let's delve specifically into the axis of symmetry, a crucial element that helps in analyzing parabolic graphs.

Graph Analysis

Analyzing the graph of a quadratic function involves identifying its key characteristics, such as the axis of symmetry, vertex, and intercepts. The vertex is the highest or lowest point on the parabola, depending on its orientation.

• The axis of symmetry is a vertical line that passes through the vertex, effectively splitting the parabola into two mirror-image halves.
• For a quadratic function \(y = ax^2 + bx + c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).
• Understanding this helps us not only locate the vertex but also provides insight into how the graph behaves.

Each quadratic function has a graph that is symmetrical along this axis. This symmetry is beneficial for solving problems related to the graph, such as determining which function shares an axis of symmetry with another.

Symmetry in Mathematics

Symmetry is a fundamental concept in mathematics, particularly evident in the study of quadratic functions. It refers to the balanced and proportionate arrangement of elements within a figure or equation. In terms of parabolas, symmetry is focused on the axis of symmetry.
Mathematically, the axis of symmetry of a parabola defined by a quadratic function \(y = ax^2 + bx + c\) guides us in understanding its geometric properties. This axis is not only a line where the graph can be "folded" over to match but also defines the vertical line around which the quadratic opens.

• Matching the axes of symmetry from different functions involves calculating \(x = -\frac{b}{2a}\) for each equation.
• In our example, matching parabolas with the symmetry \(x = -1\) confirms identical symmetrical properties despite differences in their other coefficients.
• This symmetry helps predict graph behavior and can simplify solving quadratic equations.

Understanding symmetry aids in a deeper comprehension of mathematical functions and fosters a more intuitive grasp of geometry and algebra.