Try This!
1.1 Is the equation [tex]p(x)=(x-3)(2 x+4)[/tex] in standard form or factored form?
1.2 Determine three points on the graph of [tex]p(x)=(x-3)(2 x+4)[/tex].
The points (-2, ), (0, ), and (2, ) are on the graph of p(x)!
2.1 Is the equation [tex]g(x)=-x^2-2 x+3[/tex] in standard form or factored form?
2.2 Graph [tex]g(x)[/tex] on the coordinate plane.
Answer (1)
p ( x ) = ( x − 3 ) ( 2 x + 4 ) is in factored form.
The points on the graph of p ( x ) are ( − 2 , 0 ) , ( 0 , − 12 ) , and ( 2 , − 8 ) .
g ( x ) = − x 2 − 2 x + 3 is in standard form.
The graph of g ( x ) is a parabola with vertex at ( − 1 , 4 ) , x-intercepts at ( − 3 , 0 ) and ( 1 , 0 ) , and y-intercept at ( 0 , 3 ) .
Explanation
Identifying the Form of p(x) The equation p ( x ) = ( x − 3 ) ( 2 x + 4 ) is in factored form because it is expressed as a product of its factors ( x − 3 ) and ( 2 x + 4 ) .
Calculating Points on p(x) To find the points on the graph of p ( x ) for x = − 2 , 0 , 2 , we substitute these values into the equation:
For x = − 2 :
p ( − 2 ) = ( − 2 − 3 ) ( 2 ( − 2 ) + 4 ) = ( − 5 ) ( − 4 + 4 ) = ( − 5 ) ( 0 ) = 0 So, the point is ( − 2 , 0 ) .
For x = 0 :
p ( 0 ) = ( 0 − 3 ) ( 2 ( 0 ) + 4 ) = ( − 3 ) ( 0 + 4 ) = ( − 3 ) ( 4 ) = − 12 So, the point is ( 0 , − 12 ) .
For x = 2 :
p ( 2 ) = ( 2 − 3 ) ( 2 ( 2 ) + 4 ) = ( − 1 ) ( 4 + 4 ) = ( − 1 ) ( 8 ) = − 8 So, the point is ( 2 , − 8 ) .
Identifying the Form of g(x) The equation g ( x ) = − x 2 − 2 x + 3 is in standard form because it is written in the form a x 2 + b x + c , where a = − 1 , b = − 2 , and c = 3 .
Graphing g(x) To graph g ( x ) = − x 2 − 2 x + 3 , we first find the vertex. The x-coordinate of the vertex is given by x = − 2 a b , where a = − 1 and b = − 2 .
x = − 2 ( − 1 ) − 2 = − − 2 − 2 = − 1 The y-coordinate of the vertex is found by substituting x = − 1 into g ( x ) :
g ( − 1 ) = − ( − 1 ) 2 − 2 ( − 1 ) + 3 = − 1 + 2 + 3 = 4 So, the vertex is ( − 1 , 4 ) .
Next, we find the x-intercepts by setting g ( x ) = 0 :
− x 2 − 2 x + 3 = 0 x 2 + 2 x − 3 = 0 ( x + 3 ) ( x − 1 ) = 0 So, x = − 3 or x = 1 . The x-intercepts are ( − 3 , 0 ) and ( 1 , 0 ) .
Finally, we find the y-intercept by setting x = 0 :
g ( 0 ) = − ( 0 ) 2 − 2 ( 0 ) + 3 = 3 So, the y-intercept is ( 0 , 3 ) .
We can now sketch the graph of g ( x ) using the vertex, intercepts, and a few additional points.
Examples
Understanding quadratic functions like p(x) and g(x) is essential in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown in the air or the flight of a rocket. By analyzing the equation, they can determine the maximum height reached, the range of the projectile, and the time it takes to land. Similarly, architects use quadratic functions to design curved structures like arches and bridges, ensuring stability and optimal load distribution. These functions help in optimizing designs for efficiency and safety.
