If \( f(x) = 2x^2 - 3x + 1 \), what is \( f(3) - f(-1) \)?
["# If ( f(x) = 2x^2 - 3x + 1 ), What Is ( f(3) - f(-1) )?", "Understanding how functions behave at specific values is essential in algebra and calculus. In this article, we’ll explore what happens when evaluating the quadratic function ( f(x) = 2x^2 - 3x + 1 ) at two distinct points: ( x = 3 ) and ( x = -1 ). We’ll compute ( f(3) ), find ( f(-1) ), and finally calculate the difference ( f(3) - f(-1) ).", "---", "## Step 1: Evaluate ( f(3) )", "Substitute ( x = 3 ) into the function:", "[\nf(3) = 2(3)^2 - 3(3) + 1\n]", "Calculate each term:", "- ( 2(3)^2 = 2 \ imes 9 = 18 )\n- ( -3(3) = -9 )\n- Constant term is ( +1 )", "Add them together:", "[\nf(3) = 18 - 9 + 1 = 10\n]", "---", "## Step 2: Evaluate ( f(-1) )", "Now substitute ( x = -1 ):", "[\nf(-1) = 2(-1)^2 - 3(-1) + 1\n]", "Compute each term carefully:", "- ( (-1)^2 = 1 ), so ( 2(1) = 2 )\n- ( -3(-1) = 3 )\n- Constant term: ( +1 )", "Sum the results:", "[\nf(-1) = 2 + 3 + 1 = 6\n]", "---", "## Step 3: Compute the Difference ( f(3) - f(-1) )", "Now that we have both values:", "[\nf(3) - f(-1) = 10 - 6 = 4\n]", "---", "## Why This Matters: The Function Difference Explained", "The expression ( f(3) - f(-1) ) calculates how much the function increases linearly between these two input values. In practical terms, this is useful in physics, economics, and engineering—such as modeling the area under a curve or comparing changes in systems over time.", "Understanding your functions at specific points helps not only in computation but also in interpreting behavior—whether a quadratic opens upward, touches the x-axis, or shifts vertically.", "---", "## Final Answer", "[\n\boxed{4}\n]", "So, if ( f(x) = 2x^2 - 3x + 1 ), then ( f(3) - f(-1) = 4 )."]
