To write two different quadratic functions whose graphs pass through the points (-2,0) and (4,0), we need to find two functions of the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] that satisfy these points. A quadratic function is generally determined by the form [tex]\( f(x) = a(x - x_1)(x - x_2) \)[/tex] when it passes through given points [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex].
Here, our points are [tex]\( (-2, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex]. These points represent the roots of our quadratic functions.
### Function 1
Using the factored form, we can write:
[tex]\[ f(x) = a(x + 2)(x - 4) \][/tex]
Expanding this expression, we get:
[tex]\[ f(x) = a(x^2 - 4x + 2x - 8) \][/tex]
[tex]\[ f(x) = a(x^2 - 2x - 8) \][/tex]
We have the first quadratic function:
[tex]\[ f(x) = a(x^2 - 2x - 8) \][/tex]
To create a specific function, we will let [tex]\( a = 1 \)[/tex]:
[tex]\[ f(x) = x^2 - 2x - 8 \][/tex]
### Function 2
For the second function, we can choose a different value for [tex]\( a \)[/tex]. For example, we can let [tex]\( a = 2 \)[/tex]:
[tex]\[ f(x) = 2(x + 2)(x - 4) \][/tex]
Expanding this product, we get:
[tex]\[ f(x) = 2(x^2 - 2x - 8) \][/tex]
[tex]\[ f(x) = 2x^2 - 4x - 16 \][/tex]
Thus, the second quadratic function is:
[tex]\[ f(x) = 2x^2 - 4x - 16 \][/tex]
So, the two different quadratic functions whose graphs pass through the points (-2, 0) and (4, 0) are:
1. [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
2. [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
