Graph $h(x)=0.5(x+2)^2-4$ by following these steps:
Step 1: Identify $a, h$, and $k$.
$a=$ $\square$
Answer (2)
Identify the parameters: a = 0.5 , h = − 2 , and k = − 4 from the given function h ( x ) = 0.5 ( x + 2 ) 2 − 4 .
Determine the vertex: The vertex of the parabola is ( − 2 , − 4 ) .
Create a table of values: Calculate h ( x ) for x values around the vertex.
Plot the graph: Plot the points and draw a smooth curve to represent the parabola. The value of a is 0.5 .
Explanation
Identifying Parameters a, h, and k The given function is h ( x ) = 0.5 ( x + 2 ) 2 − 4 . We need to identify the parameters a , h , and k to understand the shape and position of the parabola. The general vertex form of a parabola is given by h ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola and a determines the direction and stretch of the parabola.
Determining the Vertex Comparing the given function h ( x ) = 0.5 ( x + 2 ) 2 − 4 with the vertex form h ( x ) = a ( x − h ) 2 + k , we can identify the values of a , h , and k directly:
a = 0.5 h = − 2 k = − 4
So, the vertex of the parabola is at the point ( − 2 , − 4 ) .
Creating a Table of Values Now, let's create a table of values to plot the graph. We'll choose x values around the vertex x = − 2 to get a good representation of the parabola.
x
h ( x ) = 0.5 ( x + 2 ) 2 − 4
-4
0.5 ( − 4 + 2 ) 2 − 4 = 0.5 ( − 2 ) 2 − 4 = 0.5 ( 4 ) − 4 = 2 − 4 = − 2
-3
0.5 ( − 3 + 2 ) 2 − 4 = 0.5 ( − 1 ) 2 − 4 = 0.5 ( 1 ) − 4 = 0.5 − 4 = − 3.5
-2
0.5 ( − 2 + 2 ) 2 − 4 = 0.5 ( 0 ) 2 − 4 = 0 − 4 = − 4
-1
0.5 ( − 1 + 2 ) 2 − 4 = 0.5 ( 1 ) 2 − 4 = 0.5 ( 1 ) − 4 = 0.5 − 4 = − 3.5
0
0.5 ( 0 + 2 ) 2 − 4 = 0.5 ( 2 ) 2 − 4 = 0.5 ( 4 ) − 4 = 2 − 4 = − 2
Plotting the Graph Now we have the following points to plot:
(-4, -2) (-3, -3.5) (-2, -4) (-1, -3.5) (0, -2)
These points can be plotted on a graph to visualize the parabola.
Examples
Understanding quadratic functions like h ( x ) = 0.5 ( x + 2 ) 2 − 4 is crucial in various real-world applications. For instance, engineers use parabolas to design arches in bridges, ensuring structural stability and efficient load distribution. Similarly, in physics, the trajectory of a projectile, such as a ball thrown in the air, follows a parabolic path, which can be modeled using quadratic functions. By analyzing the parameters a , h , and k , we can predict the maximum height and range of the projectile. This knowledge is also valuable in optimizing the design of satellite dishes to focus signals at a specific point, maximizing signal strength.
The function h ( x ) = 0.5 ( x + 2 ) 2 − 4 has parameters a = 0.5 , h = − 2 , and k = − 4 , indicating its vertex at ( − 2 , − 4 ) . The parabola opens upward, and creating a table of values for x around the vertex allows for an accurate graphing of the function. By plotting these points and connecting them, we can visualize the quadratic function clearly.
;
