Question

The function \(p\) is defined as \(p(x)=x^2-3 x\). If the function \(q\) is defined as \(q(x)=p(x)-4\), what is the value of \(q(10)\) ? A) \(-30\) B) 6 C) 66 D) 70

Short answer

The value of \(q(10) = 66\). The correct answer is (C) 66.

Step-by-step solution

Find p(10)

First, substitute \(x = 10\) into the function \(p(x) = x^2 - 3x\): \(p(10) = (10)^2 - 3(10)\) Now, simply solve the equation: \(p(10) = 100 - 30 = 70\) So, \(p(10) = 70\).

Find q(10)

Now, substitute the value of \(p(10)\) into the function \(q(x) = p(x) - 4\): \(q(10) = p(10) - 4\) We already know that \(p(10) = 70\), so now we substitute it into the equation: \(q(10) = 70 - 4\) Now, simply solve the equation: \(q(10) = 66\) So, the value of \(q(10) = 66\). The correct answer is (C) 66.

Key concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, commonly taking the form \( f(x) = ax^2 + bx + c \) where \( a \) is not equal to zero. They are characterized by a curved graph known as a parabola, which can open either upwards or downwards depending on the sign of the coefficient \( a \). The highest or lowest point of the parabola is called the vertex, and the value \( x = -\frac{b}{2a} \) gives the x-coordinate of this vertex.

When solving SAT math problems involving quadratic functions, one might be asked to perform various operations such as finding the function's value for a particular \( x \) or determining its zeros, which are the \( x \) values where the graph intersects the x-axis. In the context of the original exercise, the function given, \( p(x) = x^2 - 3x \) is a quadratic function with \( a = 1 \) and \( b = -3 \) where one can apply algebraic manipulation to find specific values.

Function Substitution

Function substitution is a method used to evaluate functions at certain inputs or to express functions in terms of other functions. In practice, this involves taking a function's formula and replacing its variable with a given value or with another function's formula. The key is to perform each substitution step carefully to avoid errors.

Let's consider the original exercise. We have two functions, \( p(x) \) and \( q(x) \). The function \( q(x) \) is defined in terms of \( p(x) \)—specifically, \( q(x) = p(x) - 4 \). To find \( q(10) \), we first calculate \( p(10) \) using the definition of \( p(x) \), and then we substitute that result into the equation for \( q(x) \). This substitution step is crucial as it shows the interdependence of the two functions and how to work with composite functions where one function is defined in relation to another.

SAT Practice Problems

SAT practice problems are designed to prepare students for the types of questions they will encounter on the SAT exam. Particularly for SAT Math, these problems cover a range of topics from algebra to advanced trigonometry, and test various skills such as problem-solving, understanding of mathematical concepts, and the application of mathematical tools.

Approaching SAT practice problems methodically is essential. It's recommended to read the question carefully, identify known and unknown quantities, and choose an appropriate strategy—like function substitution, as seen in the given exercise example. It's also helpful to practice with time management, as the SAT is a timed test. By familiarizing oneself with typical SAT problem structures and practicing regularly, students can increase their speed, accuracy, and confidence, hoping to achieve a better score on the actual test.