Question

The graph of function \(g\) in the \(x y\)-plane is a parabola defined by \(g(x)=\) \((x-2)(x-4)\). Which of the following intervals contains the \(x\) coordinate of the vertex of the graph? A) \(6

Short answer

The x-coordinate of the vertex is \(3\), which is contained in the interval C) \(-2<x<4\). Hence, the correct option is C.

Step-by-step solution

Convert the function to the standard form of a parabola

Expand the given function \(g(x) = (x-2)(x-4)\) to obtain the standard form: \[g(x) = x^2 - 4x - 2x + 8\] So, the standard form of the function is: \[g(x) = x^2 - 6x + 8\]

Identify the parameters a, b, and c

From the standard form of the function \(g(x) = x^2 - 6x + 8\), we can identify the parameters as follows: \(a = 1\) \(b = -6\) \(c = 8\)

Find the x-coordinate of the vertex

Using the vertex formula \(x_v = \frac{-b}{2a}\), let's find the x-coordinate of the vertex: \[x_v = \frac{-(-6)}{2(1)}\] \[x_v = \frac{6}{2}\] \[x_v = 3\]

Identify the interval containing the x-coordinate

The x-coordinate of the vertex is \(3\). Now, let's find the interval that contains the value of 3 among the given options. A) \(6<x<8\) B) \(4<x<6\) C) \(-2<x<4\) D) \(-4<x<-2\) The correct interval is C) \(-2<x<4\), as it contains the x-coordinate of the vertex (3).

Key concepts

SAT Math Preparation

Preparing for the SAT Math section involves understanding a variety of mathematical concepts, one of which is the study of quadratic functions and their graphs, particularly parabolas. A strong grasp of how to find the vertex of a parabola is crucial, as it is a common question type on the exam.

To excel in these questions, it's important to practice converting quadratic expressions into different forms and using the vertex formula. Exercises like the one given, where students find the vertex of the parabola defined by the function \(g(x) = (x-2)(x-4)\), are excellent practice for the SAT.

Tips for SAT Math preparation include:

• Familiarizing yourself with quadratic functions and their properties.
• Understanding the vertex form of a parabola.
• Practicing a variety of problems to master these concepts.
• Reviewing mistakes in practice exercises to improve understanding and avoid similar errors on the actual test.

Practice is key, and with focused study, students can improve their ability to find the coordinates of a parabola's vertex quickly and accurately.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, and their standard form is \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants and the highest exponent of \(x\) is 2. The graph of a quadratic function is a parabola.

A parabola can open upwards or downwards, which is determined by the coefficient \(a\): if \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards. The vertex of the parabola is a significant point because it represents the maximum or minimum value of the quadratic function, depending on whether the parabola opens downwards or upwards.

Understanding the characteristics and transformations of quadratic functions is important for analyzing their graphs and for finding the vertex, which is the turning point of the parabola. The process of converting the factored form, such as \((x-2)(x-4)\), into standard form is a fundamental skill in algebra and helps to reveal the properties of the quadratic function, such as the vertex and the y-intercept.

Vertex Form of a Parabola

The vertex form of a quadratic function is particularly useful when dealing with parabolas, as it makes it easy to identify the vertex. The vertex form is given by \(f(x) = a(x-h)^2 + k\), where \((h, k)\) are the coordinates of the vertex, and \(a\) is the coefficient that determines the direction of the parabola's opening.

By analyzing the vertex form, one can immediately find the vertex of the parabola, which, again, is the highest or lowest point on the graph depending on whether the parabola opens upwards or downwards. Furthermore, the vertex form is instrumental in solving vertex-related problems and in graphing parabolas.

To convert a quadratic function from standard form to vertex form, one uses a process called completing the square. However, in some cases, like the one on the SAT exercise, the vertex can be found without completing the square, by using the formula \(x_v = \frac{-b}{2a}\), derived from the standard form coefficients. Knowing how to manipulate these forms is a valuable skill not only for SAT preparation but also for higher-level mathematics.