Question

the table of values for function $f(x)$ is shown.

$x$	$f(x)$
$-3$	$-2$
$0$	$-5$
$3$	$-2$
$6$	$1$

what type of function is $f(x)$?

Explanation:

Step1: Analyze symmetry

Check if \( f(-x) = f(x) \) (even) or \( f(-x)=-f(x) \) (odd). For \( x = 6 \) (not in table, but \( x = - 6 \) and \( x = 6 \)): \( f(-6)=1 \), \( f(6)=1 \); \( x=-3 \), \( f(-3)=-2 \); \( x = 3 \), \( f(3)=-2 \); \( x = 0 \), \( f(0)=-5 \), \( f(-0)=f(0) \). So symmetric about y - axis? Wait, check the values. Let's list pairs: \( (-6,1) \) and \( (6,1) \), \( (-3,-2) \) and \( (3,-2) \), \( (0,-5) \). This is a parabola - like symmetry, quadratic function (even function, symmetric about y - axis). Also, check the differences. First, find the first differences (change in \( f(x) \) for change in \( x \) of 3): from \( x=-6 \) to \( x=-3 \), \( \Delta x = 3 \), \( \Delta f(x)=-2 - 1=-3 \); \( x=-3 \) to \( x = 0 \), \( \Delta x = 3 \), \( \Delta f(x)=-5-(-2)=-3 \); \( x = 0 \) to \( x = 3 \), \( \Delta x = 3 \), \( \Delta f(x)=-2-(-5)=3 \); \( x = 3 \) to \( x = 6 \), \( \Delta x = 3 \), \( \Delta f(x)=1-(-2)=3 \). Now second differences: from first \( \Delta f=-3 \) to next \( \Delta f=-3 \), difference is 0; then from \( \Delta f=-3 \) to \( \Delta f = 3 \), difference is 6; from \( \Delta f = 3 \) to \( \Delta f = 3 \), difference is 0? Wait, no, maybe my interval is wrong. Wait, the x - values are spaced by 3. Let's consider the function. The symmetry about the y - axis (since \( f(-x)=f(x) \): \( f(-6)=f(6)=1 \), \( f(-3)=f(3)=-2 \), \( f(0)=f(0) \)) suggests it's an even function, which is characteristic of a quadratic function (parabola opening up or down, symmetric about y - axis when the vertex is on y - axis). Also, the second differences (for a quadratic function, second differences are constant). Let's re - calculate differences with \( \Delta x = 3 \). Let's list \( x \): - 6, - 3, 0, 3, 6; \( f(x) \): 1, - 2, - 5, - 2, 1. First differences (between consecutive \( f(x) \)): - 2 - 1=-3; - 5-(-2)=-3; - 2-(-5)=3; 1-(-2)=3. Now second differences (between first differences): - 3-(-3)=0; 3-(-3)=6; 3 - 3 = 0. Wait, that's not constant. Wait, maybe the step is 3, but for a quadratic function \( f(x)=ax^{2}+bx + c \). Since \( f(-x)=f(x) \), \( b = 0 \). So \( f(x)=ax^{2}+c \). Let's plug in \( x = 0 \): \( f(0)=c=-5 \). So \( f(x)=ax^{2}-5 \). Plug in \( x = 3 \): \( f(3)=9a-5=-2 \) → \( 9a=3 \) → \( a=\frac{1}{3} \). So \( f(x)=\frac{1}{3}x^{2}-5 \). Let's check \( x = 6 \): \( f(6)=\frac{1}{3}(36)-5 = 12 - 5 = 7 \)? Wait, no, the table says \( f(6)=1 \). Wait, maybe my assumption is wrong. Wait, \( x = 6 \): \( f(6)=1 \). If \( f(x)=ax^{2}+bx + c \), and \( f(-x)=f(x) \), so \( b = 0 \). Then \( f(6)=36a + c = 1 \), \( f(3)=9a + c=-2 \), \( f(0)=c=-5 \). From \( f(0)=c=-5 \), then \( 9a-5=-2 \) → \( 9a = 3 \) → \( a=\frac{1}{3} \). Then \( f(6)=\frac{1}{3}(36)-5=12 - 5 = 7 \), but table says 1. Oh, so maybe the function is symmetric about the y - axis but with a different form? Wait, the values at \( x=-6,6 \) are 1; \( x=-3,3 \) are - 2; \( x = 0 \) is - 5. Let's see the pattern: \( x^{2} \) values: 36, 9, 0, 9, 36. Let's see \( f(x) \) in terms of \( x^{2} \). Let \( x^{2}=36 \), \( f(x)=1 \); \( x^{2}=9 \), \( f(x)=-2 \); \( x^{2}=0 \), \( f(x)=-5 \). The difference between \( x^{2} \) and \( f(x) \): 36 - 1 = 35; 9-(-2)=11; 0 - (-5)=5. No, that's not linear. Wait, the first differences when \( x \) increases by 3: from - 6 to - 3 (x increases by 3), f(x) decreases by 3; - 3 to 0, decreases by 3; 0 to 3, increases by 3; 3 to 6, increases by 3. So the graph is a parabola (quadratic function) because it's symmetric about the y - axis (even function) and the shape of the values (symmetric, U - shaped or inverted U - shaped? Wait, from x=-6 to 0, f(x) decreases: 1, - 2, - 5; from x = 0 to 6, f(x) increases: - 5, - 2, 1. So it's a U - shaped curve, symmetric about x = 0 (y - axis), so it's a quadratic function (parabola), which is a type of polynomial function, specifically quadratic (degree 2). So the function is a quadratic function (or even function, symmetric about y - axis, which is characteristic of quadratic functions with \( b = 0 \)).

Step2: Conclusion

Since the function values are symmetric about the y - axis ( \( f(-x)=f(x) \) for \( x=-6, - 3,0,3,6 \)) and the graph of the function (from the table) has a U - shape (decreasing then increasing) with symmetry about the y - axis, this is a quadratic function (a type of polynomial function of degree 2). Also, the second differences (when considering the correct interval or the symmetry) indicate a quadratic relationship.

Answer:

The function \( f(x) \) is a quadratic function (or a parabola - shaped function, symmetric about the y - axis, an even function). Specifically, it is a quadratic function.