91Ó°ÊÓ

Evaluating quadratic functions involves substituting a given value into the equation to find the output. The standard form of a quadratic function is typically written as:

\[ f(x) = ax^2 + bx + c \]
In our specific case, the function given is \[ f(x) = 2x^2 - x + 3. \] A quadratic function has the variable raised to the second power, making its graph a parabola.
Below are the steps to evaluate the function for given values of x:

• Identify the value of x you want to substitute.
• Replace every instance of x in the function with the given value.
• Perform the arithmetic operations.

Using these steps, you can find the output of the quadratic function for any input value.

Substitution is a common method used to find the value of a function at a given point. It involves replacing the variable in the function with the specified number or variable. Here's how it works for the given function \[ f(x) = 2x^2 - x + 3: \]
1. To find f(2), substitute x = 2:
\[ f(2) = 2(2)^2 - 2 + 3 \]
Calculate the power:\[ 2^2 = 4 \]Multiply by 2: \[ 2(4) = 8 \]Now perform the remaining operations:\[ 8 - 2 + 3 = 9 \]So,\[ f(2) = 9. \]2. To find f(-1), substitute x = -1:
\[ f(-1) = 2(-1)^2 - (-1) + 3 \]Calculate the power:\[ (-1)^2 = 1 \]Multiply by 2: \[ 2(1) = 2 \]Now perform the remaining operations:\[ 2 + 1 + 3 = 6 \]So, \[ f(-1) = 6. \]3. To find f(a), substitute x = a:
\[ f(a) = 2a^2 - a + 3 \]Since a is a variable, \[ f(a) \] cannot be simplified further.
Using substitution helps you evaluate functions accurately and efficiently.

When evaluating functions, the variable acts as a placeholder that can be replaced by numbers or other variables. This process helps to understand how the function behaves with different inputs. Here, we consider an example function \[ f(x) = 2x^2 - x + 3. \]

• Given Numbers: When evaluating for specific numbers, like f(2) or f(-1), substitute the numbers into the function and simplify.
• Other Variables: When given another variable (like f(a)), substitute the variable into the function. This is useful for forming a more generalized understanding of the function.

To evaluate the function, replace x with the given number or variable:
1. If \[ x = 2, f(2) = 9. \]
2. If \[ x = -1, f(-1) = 6. \]
3. If \[ x = a, f(a) = 2a^2 - a + 3. \]Understanding variable evaluation helps you manage various scenarios in functions. Whether you work with numbers or variables, you can systematically determine the output.