Question Statement

Two asteroids are following parabolic paths represented by the functions: $f (x) = x^{2} - 7 x + 12$ $g (x) = x (x - 3)$

By drawing the graphs of both trajectories, find the coordinates of the location where both asteroids will pass (the point of intersection).

Background and Explanation

To find where two trajectories meet, we look for the intersection points of their graphs. This occurs at the $(x, y)$ coordinates that satisfy both equations simultaneously. In this problem, we use a table of values to plot the parabolas and identify the shared point.

Solution

To find the intersection point, we first evaluate both functions across a range of $x$ values to understand their shapes and identify where their outputs match.

Step 1: Tabulate values for both functions

Using the provided data, we calculate the $y$ -coordinates for $f (x)$ and $g (x)$ :

$x$	-1	0	1	1.5	2	3	3.5	4	5
$f (x)$	20	12	6	3.75	2	0	-0.25	0	2
$g (x)$	4	0	-2	-2.25	-2	0	1.75	4	10

Step 2: Analyze the results

By observing the table, we look for an $x$ value where $f (x) = g (x)$ .

At $x = 3$ :
- $f (3) = 3^{2} - 7 (3) + 12 = 9 - 21 + 12 = 0$
- $g (3) = 3 (3 - 3) = 3 (0) = 0$

Since both functions yield a value of $0$ when $x$ is $3$ , the point $(3, 0)$ is a point of intersection.

Step 3: Algebraic Verification

To ensure there are no other intersection points not captured by the table, we can set the two equations equal to each other: $x^{2} - 7 x + 12 = x (x - 3)$ $x^{2} - 7 x + 12 = x^{2} - 3 x$

Subtract $x^{2}$ from both sides: $- 7 x + 12 = - 3 x$

Add $7 x$ to both sides: $12 = 4 x$ $x = 3$

Substituting $x = 3$ back into $g (x)$ : $g (3) = 3 (3 - 3) = 0$

The asteroids will both pass through the point $(3, 0)$ .

Key Formulas or Methods Used

Quadratic Function Form: $f (x) = a x^{2} + b x + c$
Point-Plotting Method: Creating a table of $(x, y)$ pairs to draw a graph.
Intersection of Graphs: Solving $f (x) = g (x)$ to find where two paths cross.

Summary of Steps

List the equations for both parabolic paths.
Create a table of values for $x$ ranging from -1 to 5.
Calculate the corresponding $y$ values for both $f (x)$ and $g (x)$ .
Identify the $x$ value where both functions produce the same $y$ result.
Conclude that the intersection point is $(3, 0)$ .

x

-1

1.5

3.5

$f (x)$

3.75

-0.25

$g (x)$

-2

-2.25

-2

1.75

Step 3: Algebraic Verification

To ensure there are no other intersection points not captured by the table, we can set the two equations equal to each other:

x^{2} - 7 x + 12 = x (x - 3)

x^{2} - 7 x + 12 = x^{2} - 3 x

Subtract

x^{2}

from both sides:

- 7 x + 12 = - 3 x

Add

7 x

to both sides:

12 = 4 x

x = 3

Substituting

x = 3

back into

g (x)

g (3) = 3 (3 - 3) = 0

The asteroids will both pass through the point $(3, 0)$ .

1.2 Q-11

Question Statement

Background and Explanation

Solution

Step 1: Tabulate values for both functions

Step 2: Analyze the results

Step 3: Algebraic Verification

Key Formulas or Methods Used

Summary of Steps

1.2 Q-11

Question Statement

Background and Explanation

Solution

Step 1: Tabulate values for both functions

Step 2: Analyze the results

Step 3: Algebraic Verification

Key Formulas or Methods Used

Summary of Steps