Graph the function.
$h(x)=-\frac{1}{3} x^2+2 x-4$
Answer (1)
Find the vertex of the parabola using x = − 2 a b and h ( x ) . The vertex is ( 3 , − 1 ) .
Determine if the function has real roots by calculating the discriminant. Since the discriminant is negative, there are no real roots.
Find the y-intercept by setting x = 0 . The y-intercept is ( 0 , − 4 ) .
Sketch the graph of the parabola using the vertex and y-intercept. h ( x ) = − 3 1 x 2 + 2 x − 4
Explanation
Analyze the Function We are given the quadratic function h ( x ) = − 3 1 x 2 + 2 x − 4 and asked to graph it. Since the coefficient of the x 2 term is negative, the parabola opens downward, indicating that it has a maximum point.
Find the Vertex x-coordinate To graph the function, we first need to find the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b , where a = − 3 1 and b = 2 . Thus, x = − 2 ( − 3 1 ) 2 = − − 3 2 2 = 3.
Find the Vertex y-coordinate Now, we find the y-coordinate of the vertex by plugging x = 3 into the function: h ( 3 ) = − 3 1 ( 3 ) 2 + 2 ( 3 ) − 4 = − 3 1 ( 9 ) + 6 − 4 = − 3 + 6 − 4 = − 1. So, the vertex of the parabola is ( 3 , − 1 ) .
Check for Real Roots Next, we determine if the function has any real roots. The discriminant is given by Δ = b 2 − 4 a c . In this case, a = − 3 1 , b = 2 , and c = − 4 . So, Δ = ( 2 ) 2 − 4 ( − 3 1 ) ( − 4 ) = 4 − 3 16 = 3 12 − 16 = − 3 4 . Since the discriminant is negative, the function has no real roots, meaning the parabola does not intersect the x-axis.
Find Additional Points To get a better understanding of the graph, we can find a couple of additional points. Let's find the y-intercept by setting x = 0 : h ( 0 ) = − 3 1 ( 0 ) 2 + 2 ( 0 ) − 4 = − 4. So, the y-intercept is ( 0 , − 4 ) . Since the parabola is symmetric about the vertical line through the vertex, we can find another point by reflecting the y-intercept across the line x = 3 . The y-intercept is 3 units to the left of the vertex, so the reflected point will be 3 units to the right of the vertex, which is x = 3 + 3 = 6 . The y-coordinate will be the same, so the point is ( 6 , − 4 ) .
Sketch the Graph Now we can sketch the graph. The parabola opens downward, has a vertex at ( 3 , − 1 ) , a y-intercept at ( 0 , − 4 ) , and passes through the point ( 6 , − 4 ) .
Final Answer The vertex of the function is ( 3 , − 1 ) . The function has no real roots. The y-intercept is ( 0 , − 4 ) .
Examples
Understanding quadratic functions is crucial in various real-world applications, 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 that focus light or radio waves to a single point. Economists use quadratic functions to model cost and revenue curves to optimize profits.
