Question

Sanjay graphs a quadratic function that has x-intercepts of –3 and 7. Which functions could he have graphed? Check all that apply. g(x) = x2 – 4x – 21 g(x) = (x – 3)(x + 7) g(x) = 3x2 – 12x – 63 g(x) = –(x + 3)(x – 7) g(x) = x2 + 4x – 21

Answer 1

X-intercepts are found by factoring.  Use the quadratic formula on the first since it's in standard form and you find that your x values are in fact -3 and 7.  For the second one, use the Zero Product Property that says that x - 3 = 0 or x + 7 = 0.  Therefore, x = 3 and -7.  Signs are wrong.  So not the second one.  As for the third one, if you factor out a 3, your polynomial is exactly the same as the first one which did give us the desired x values.  So the third one checks out.  If you FOIL out the first one and then apply the quadratic formula you do get x = 3 and -7. So the fourth one checks out too.  For the last one, putting it into the quadratic formula gives you x values of 3 and -7, so no to that one.  Summary:  1st, 3rd, 4th have zeros of -3 and 7; 2nd and 5th do not.

Answer 2

Answer: The functions could be,

g(x) = x² - 4x - 21,

g(x) = 3x² - 12x - 63,

g(x) = -(x + 3)(x - 7)

Step-by-step explanation:

First option,

$g(x) = x^2-4x-21$,

For x-intercept, g(x) = 0,

$\implies x^2-4x-21=0$

$x^2-7x+3x-21=0$

$x(x-7)+3(x-7)=0$

$(x+3)(x-7)=0$

$\implies x=-3,7$

⇒ The x-intercept of g(x) = x² - 4x - 21 are -3 and 7.

Second option,

$g(x) = (x-3)(x+7)$,

For x-intercept, g(x) = 0,

$(x-3)(x+7)=0$

$\implies x=3,-7$

⇒ The x-intercept of g(x) = (x-3)(x+7) are 3 and -7.

Third option,

$g(x) = 3x^2-12x-63$,

For x-intercept, g(x) = 0,

$g(x) = 3x^2-12x-63=0$,

$\implies x^2-4x-21=0$

$(x+3)(x-7)=0$

$\implies x=-3,7$

⇒ The x-intercept of g(x) = 3x² - 12x - 63 are -3 and 7.

Fourth option,

$g(x) = -(x+3)(x-7)$,

For x-intercept, g(x) = 0,

$-(x+3)(x-7)=0$

$\implies x=-3,7$

⇒ The x-intercept of g(x) = -(x+3)(x-7) are -3 and 7.

Fifth option,

$g(x) = x^2+4x-21$,

For x-intercept, g(x) = 0,

$\implies x^2+4x-21=0$

$x^2+7x-3x-21=0$

$x(x+7)-3(x+7)=0$

$(x-3)(x+7)=0$

$\implies x=3,-7$

⇒ The x-intercept of g(x) = x² + 4x - 21 are 3 and -7.