Question

1. function f and g are shown in the coordinate plane below. the parabola that represents function f passes through the points b and c. the parabola that represents function g passes through points a and b. point a is the vertex of function g. point b is located on the y - axis. point c is located on the x - axis. the x - coordinate of point a is the same as the x - coordinate of point c. point a is 14 units below point c. if the rule of function f is f(x) = - 0.5x² + 6x - 10, what is the rule of function g?

f(x) = - 0.5x² + 6x - 10
0 = - 0.5x² + 6x - 10
x = \frac{ - b \pm \sqrt{b² - 4ac}}{2a} = \frac{ - 6 \pm \sqrt{6² - 4( - 0.5)( - 10)}}{2( - 0.5)} = \frac{ - 6 \pm \sqrt{36 - 20}}{ - 1} = \frac{ - 6 \pm \sqrt{16}}{ - 1} = \frac{ - 6 \pm 4}{ - 1}
2 - 14 = - 12

Explanation:

the x - coordinate of A is the same as that of C, and A is 14 units below C (y - coordinate of C is 0, so y - coordinate of A is \( 0 - 14=-14 \)). So if C is \( (2,0) \), A is \( (2,-14) \); if C is \( (10,0) \), A is \( (10,-14) \).

Step 4: Find the equation of \( g(x) \) using vertex form

The vertex form of a parabola is \( g(x)=a(x - h)^2+k \), where \( (h,k) \) is the vertex (point A). For \( A=(2,-14) \), \( g(x)=a(x - 2)^2-14 \). Since \( g(x) \) passes through \( B=(0,-10) \), substitute \( x = 0 \), \( y=-10 \):

$$ -10=a(0 - 2)^2-14 $$

$$ -10 = 4a-14 $$

$$ 4a=4\Rightarrow a = 1 $$

So \( g(x)=(x - 2)^2-14=x^2-4x + 4-14=x^2-4x - 10 \).
For \( A=(10,-14) \), \( g(x)=a(x - 10)^2-14 \), substituting \( x = 0 \), \( y=-10 \) gives \( a = 0.04 \), which is less likely from the graph (parabola \( g \) is narrower, so \( a = 1 \) is better).

Answer:

\( g(x)=x^2-4x - 10 \)