Graph the function.
[tex]y=\frac{1}{2} x^2+2 x-8[/tex]
Answer (1)
Find the vertex of the parabola using x = − 2 a b and substitute to find the y-coordinate: Vertex is ( − 2 , − 10 ) .
Find the x-intercepts by setting y = 0 and using the quadratic formula: x ≈ − 6.47 and x ≈ 2.47 .
Find the y-intercept by setting x = 0 : y = − 8 .
Sketch the parabola using the vertex and intercepts. The final answer is a parabola with vertex ( − 2 , − 10 ) , x-intercepts approximately ( − 6.47 , 0 ) and ( 2.47 , 0 ) , and y-intercept ( 0 , − 8 ) .
Explanation
Understanding the Problem We are given the quadratic function y = 2 1 x 2 + 2 x − 8 and asked to graph it. This is a parabola. To graph it, we need to find the vertex, x-intercepts, and y-intercept.
Finding the Vertex First, let's find the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b , where a = 2 1 and b = 2 . So, x = − 2 ( 2 1 ) 2 = − 1 2 = − 2 . To find the y-coordinate of the vertex, we substitute x = − 2 into the equation: y = 2 1 ( − 2 ) 2 + 2 ( − 2 ) − 8 = 2 1 ( 4 ) − 4 − 8 = 2 − 4 − 8 = − 10 . Thus, the vertex is ( − 2 , − 10 ) .
Finding the X-Intercepts Next, let's find the x-intercepts by setting y = 0 and solving for x : 2 1 x 2 + 2 x − 8 = 0 . Multiplying by 2 to eliminate the fraction, we get x 2 + 4 x − 16 = 0 . We can use the quadratic formula to solve for x : x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = 4 , and c = − 16 . So, x = 2 ( 1 ) − 4 ± 4 2 − 4 ( 1 ) ( − 16 ) = 2 − 4 ± 16 + 64 = 2 − 4 ± 80 = 2 − 4 ± 4 5 = − 2 ± 2 5 . Thus, the x-intercepts are approximately − 2 − 2 5 ≈ − 6.47 and − 2 + 2 5 ≈ 2.47 .
Finding the Y-Intercept Now, let's find the y-intercept by setting x = 0 : y = 2 1 ( 0 ) 2 + 2 ( 0 ) − 8 = − 8 . Thus, the y-intercept is ( 0 , − 8 ) .
Sketching the Parabola We have the vertex ( − 2 , − 10 ) , x-intercepts approximately ( − 6.47 , 0 ) and ( 2.47 , 0 ) , and the y-intercept ( 0 , − 8 ) . We can now sketch the parabola using these points. The parabola opens upwards since the coefficient of x 2 is positive.
Final Answer The graph of the function y = 2 1 x 2 + 2 x − 8 is a parabola with vertex at ( − 2 , − 10 ) , x-intercepts at approximately ( − 6.47 , 0 ) and ( 2.47 , 0 ) , and y-intercept at ( 0 , − 8 ) .
Examples
Understanding quadratic functions is crucial in various fields, such as physics, engineering, and economics. For instance, the trajectory of a projectile, like a ball thrown in the air, can be modeled by a quadratic function. By finding the vertex of the parabola, we can determine the maximum height the ball reaches. Similarly, engineers use quadratic functions to design parabolic mirrors and antennas, where the focus point is essential for optimal performance. Economists also use quadratic functions to model cost and revenue curves, helping businesses optimize their production and pricing strategies. In essence, mastering quadratic functions provides a powerful tool for analyzing and solving real-world problems across multiple disciplines.
