4.1 Simplifying Expressions and Evaluating Functions

The Role of Algebra on the SHSAT

Algebra represents a significant portion of the SHSAT Math section. Scoring highly requires not just memorizing rules, but understanding the underlying mechanics of algebraic terms, properties, and relationships. Section 4.1 focuses on the building blocks of algebra: algebraic expressions, simplification techniques, and basic function notation. These skills are essential for both pure algebra questions and setting up variables in word problems.

Key Algebraic Definitions

Before diving into operations, you must master the precise vocabulary of algebra. The SHSAT often tests your understanding of these terms indirectly through word problems. For instance, a problem might ask you to find the sum of coefficients in a simplified expression. If you do not know what a coefficient is, you cannot answer the question even if you know how to simplify it.

• Variable: A symbol (usually a letter like $x$, $y$, or $a$) that represents an unknown number.
• Constant: A fixed number that does not change in value (e.g., $5$, $-12$, $\frac{2}{3}$).
• Coefficient: The numerical factor multiplied by a variable (e.g., in $7x$, the coefficient is $7$; in $-y$, the coefficient is $-1$).
• Term: A single number, variable, or product of numbers and variables separated by addition or subtraction operators (e.g., in the expression $3x^2 - 5x + 8$, the terms are $3x^2$, $-5x$, and $8$).
• Expression: A mathematical phrase that can contain numbers, variables, and operators, but does not contain an equal sign (e.g., $4x + 7$). An expression cannot be "solved," only simplified or evaluated.

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Simplifying Expressions and Combining Like Terms

To simplify an algebraic expression, you must combine "like terms." Like terms are terms that have the exact same variables raised to the exact same exponents. For example, $3x$ and $5x$ are like terms, while $3x$ and $3x^2$ are unlike terms because the exponents do not match. Similarly, $2xy$ and $5yx$ are like terms because multiplication is commutative, meaning the order of variables does not matter.

When combining like terms, you add or subtract their coefficients while leaving the variable part completely unchanged. This is similar to saying: 3 apples + 5 apples = 8 apples. In mathematical terms: $7x^2 - 3x^2 = (7 - 3)x^2 = 4x^2$ Do not add the exponents when combining like terms. A common student error is writing $7x^2 - 3x^2 = 4x^0$ or $4$. The exponent remains exactly the same.

The Distributive Property

The Distributive Property allows you to multiply a single term by two or more terms inside a set of parentheses. It is represented by the formula: $a(b + c) = ab + ac$ On the SHSAT, the most common trap involves distributing a negative coefficient. When distributing a negative number, you must reverse the sign of every term inside the parentheses.

• Example: Simplify $-3(2x - 4)$. Distribute $-3$ to $2x$: $-3 \times 2x = -6x$. Distribute $-3$ to $-4$: $-3 \times (-4) = +12$. Simplified expression: -6x + 12.

Let's look at a multi-step simplification problem that combines both distributing and combining like terms: Simplify the expression: $4(2x - 3y) - 2(x - 5y) + 7$.

1. Distribute the first coefficient: Multiply $4$ by both $2x$ and $-3y$: $4(2x - 3y) = 8x - 12y$
2. Distribute the second coefficient: Multiply $-2$ by both $x$ and $-5y$. Be extremely careful with the negative sign! Multiplying a negative by a negative yields a positive: $-2(x - 5y) = -2x + 10y$
3. Combine the terms: Write out the full expression and group like terms together: $8x - 12y - 2x + 10y + 7$ $(8x - 2x) + (-12y + 10y) + 7$
4. Final Simplification: Perform the addition and subtraction on the coefficients: $6x - 2y + 7$

Common Algebra Pitfalls to Avoid on the SHSAT

• Forgetting to distribute to the second term: In the expression $5(x + 2)$, a common error is writing $5x + 2$. Make sure you multiply the $5$ by both the $x$ and the $2$ to get $5x + 10$.
• Improper signs when distributing negatives: When simplifying $-(x - 3)$, the negative sign outside the parentheses acts as a $-1$. Distributing it yields $-x + 3$, not $-x - 3$.
• Combining unlike terms: You cannot combine $3x$ and $4y$ to get $7xy$. They must remain separate. Likewise, $2x^2 + 3x$ cannot be simplified further.

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Evaluating Algebraic Expressions

Evaluating an expression means finding its numerical value by substituting given numbers for the variables. The key to evaluating expressions without error is using parentheses during substitution, especially for negative numbers.

The Exponent Trap with Negatives

A frequent source of errors on the SHSAT is evaluating squared negative numbers.

• Evaluate $-x^2$ when $x = -3$. Substitute $-3$ inside parentheses for the variable $x$: $-(-3)^2$. Following the Order of Operations (PEMDAS), evaluate the exponent first: $(-3)^2 = 9$. Then apply the negative sign: $-(9) = -9$.
• Evaluate $(-x)^2$ when $x = -3$. Substitute $-3$ inside: $(-(-3))^2 = (3)^2 = 9$. Notice how the placement of parentheses changes the result. Always substitute variables within parentheses to maintain structural integrity.

Complex Substitution Example

Let's evaluate a more complex fractional algebraic expression: Evaluate the expression $\frac{3a^2 - 2b}{c + a}$ for $a = -2$, $b = 3$, $c = -1$.

1. Substitute values with parentheses: $\frac{3(-2)^2 - 2(3)}{(-1) + (-2)}$
2. Evaluate the numerator: First, compute the exponent: $(-2)^2 = 4$. Next, multiply: $3(4) - 2(3) = 12 - 6 = 6$.
3. Evaluate the denominator: Combine the negative numbers: $-1 + (-2) = -3$.
4. Divide numerator by denominator: $\frac{6}{-3} = -2$ The final value of the expression is -2. Keeping track of signs in fractions is crucial.

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Introduction to Function Notation

A function is a mathematical relationship where each input (domain value) corresponds to exactly one output (range value). In algebra, functions are written in function notation, typically as $f(x)$, read as "f of x."

It is vital to understand that $f(x)$ does not mean $f$ multiplied by $x$. Instead, $f$ is the name of the function, $x$ is the input value, and $f(x)$ is the output value. Graphically, the output $f(x)$ corresponds to the $y$-value.

Evaluating Functions

To evaluate a function at a specific value, you substitute that value for $x$ everywhere it appears in the function's equation.

• Example: If $f(x) = x^2 - 4x + 5$, find $f(-2)$. Substitute $-2$ for $x$: $f(-2) = (-2)^2 - 4(-2) + 5$ $f(-2) = 4 + 8 + 5 = 17$

On the SHSAT, you may also see equations involving multiple functions, such as finding $f(3) + g(2)$, or equations with parameters.

• Example: If $f(x) = 3x - 1$ and $g(x) = x^2$, find $2f(4) - g(-3)$. First, find $f(4)$: $f(4) = 3(4) - 1 = 11$. Second, find $g(-3)$: $g(-3) = (-3)^2 = 9$. Now substitute these values into the expression: $2(11) - 9 = 22 - 9 = 13$.

Functions as Real-World Processes

Functions can represent real-world scenarios. For example, a taxi ride's cost can be modeled by the function: $C(m) = 2.50 + 1.80m$ where $m$ is the number of miles traveled, and $C(m)$ is the total cost in dollars. If a passenger travels $10$ miles, the cost is found by evaluating $C(10)$: $C(10) = 2.50 + 1.80(10) = 2.50 + 18.00 = 20.50$ So, the ride costs 20.50. This application of functions connects abstract algebra to real-world word problems.