Question

write a quadratic function to represent the values in the table.
x | -1 | 0 | 1 | 2 | 3
y | 13 | 3 | -3 | -5 | -3
f(x) =
(use integers or decimals for any numbers in the expression. type your answer in standard form.)

Explanation:

Step1: Recall quadratic function form

The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \). We can use the given points \((x, y)\) from the table to set up a system of equations. The points are: when \( x = -1 \), \( y = 13 \); \( x = 0 \), \( y = 3 \); \( x = 1 \), \( y = -3 \); \( x = 2 \), \( y = -5 \); \( x = 3 \), \( y = -3 \). Let's start with \( x = 0 \), \( y = 3 \). Substituting into \( f(x) = ax^2 + bx + c \), we get \( 3 = a(0)^2 + b(0) + c \), so \( c = 3 \).

Step2: Use \( x = 1 \), \( y = -3 \)

Now we know \( c = 3 \), substitute \( x = 1 \), \( y = -3 \) and \( c = 3 \) into \( f(x) = ax^2 + bx + c \): \( -3 = a(1)^2 + b(1) + 3 \), which simplifies to \( a + b + 3 = -3 \), so \( a + b = -6 \) (Equation 1).

Step3: Use \( x = -1 \), \( y = 13 \)

Substitute \( x = -1 \), \( y = 13 \) and \( c = 3 \) into \( f(x) = ax^2 + bx + c \): \( 13 = a(-1)^2 + b(-1) + 3 \), which simplifies to \( a - b + 3 = 13 \), so \( a - b = 10 \) (Equation 2).

Step4: Solve the system of equations

Now we have Equation 1: \( a + b = -6 \) and Equation 2: \( a - b = 10 \). Add the two equations together: \( (a + b) + (a - b) = -6 + 10 \), which gives \( 2a = 4 \), so \( a = 2 \). Substitute \( a = 2 \) into Equation 1: \( 2 + b = -6 \), so \( b = -8 \).

Step5: Write the quadratic function

Now we have \( a = 2 \), \( b = -8 \), \( c = 3 \). Substitute these into the standard form: \( f(x) = 2x^2 - 8x + 3 \). We can check with other points, for example, \( x = 2 \): \( f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5 \), which matches the table. \( x = 3 \): \( f(3) = 2(9) - 8(3) + 3 = 18 - 24 + 3 = -3 \), which also matches.

Answer:

\( f(x) = 2x^2 - 8x + 3 \)