Graph the quadratic function by plotting the vertex and some intercepts: [tex]$f(x)=4 x^2-8 x+3$[/tex]
Answer (1)
Find the vertex of the parabola using x v = − 2 a b and f ( x v ) , which gives ( 1 , − 1 ) .
Determine the y-intercept by setting x = 0 , resulting in ( 0 , 3 ) .
Calculate the x-intercepts by solving 4 x 2 − 8 x + 3 = 0 , yielding ( 2 1 , 0 ) and ( 2 3 , 0 ) .
Plot these points to graph the quadratic function f ( x ) = 4 x 2 − 8 x + 3 . The final answer is vertex ( 1 , − 1 ) , y-intercept ( 0 , 3 ) , and x-intercepts ( 2 1 , 0 ) and ( 2 3 , 0 ) .
vertex ( 1 , − 1 ) , x-intercepts ( 0.5 , 0 ) and ( 1.5 , 0 ) , y-intercept ( 0 , 3 )
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = 4 x 2 − 8 x + 3 and asked to graph it by finding the vertex and some intercepts. This involves finding the x-intercepts (where the function equals zero), the y-intercept (where x equals zero), and the vertex (the minimum or maximum point of the parabola).
Finding the Vertex x-coordinate The vertex of a quadratic function in the form f ( x ) = a x 2 + b x + c is given by the point ( x v , f ( x v )) , where x v = − 2 a b . In our case, a = 4 and b = − 8 . Thus, the x-coordinate of the vertex is: x v = − 2 ( 4 ) − 8 = 8 8 = 1
Finding the Vertex y-coordinate Now we find the y-coordinate of the vertex by plugging x v = 1 into the function: f ( 1 ) = 4 ( 1 ) 2 − 8 ( 1 ) + 3 = 4 − 8 + 3 = − 1 So, the vertex is at the point ( 1 , − 1 ) .
Finding the y-intercept To find the y-intercept, we set x = 0 in the function: f ( 0 ) = 4 ( 0 ) 2 − 8 ( 0 ) + 3 = 0 − 0 + 3 = 3 Thus, the y-intercept is at the point ( 0 , 3 ) .
Finding the x-intercepts To find the x-intercepts, we set f ( x ) = 0 and solve for x :
4 x 2 − 8 x + 3 = 0 We can use the quadratic formula to solve for x : x = 2 a − b ± b 2 − 4 a c . In this case, a = 4 , b = − 8 , and c = 3 . So, x = 2 ( 4 ) 8 ± ( − 8 ) 2 − 4 ( 4 ) ( 3 ) = 8 8 ± 64 − 48 = 8 8 ± 16 = 8 8 ± 4 The two x-intercepts are: x 1 = 8 8 + 4 = 8 12 = 2 3 = 1.5 x 2 = 8 8 − 4 = 8 4 = 2 1 = 0.5 So, the x-intercepts are ( 2 1 , 0 ) and ( 2 3 , 0 ) .
Graphing the Function We have found the vertex ( 1 , − 1 ) , the y-intercept ( 0 , 3 ) , and the x-intercepts ( 2 1 , 0 ) and ( 2 3 , 0 ) . We can now plot these points and draw a smooth curve through them to graph the quadratic function. The parabola opens upwards since the coefficient of x 2 is positive ( 0"> a = 4 > 0 ).
Final Answer The vertex of the quadratic function is ( 1 , − 1 ) , the y-intercept is ( 0 , 3 ) , and the x-intercepts are ( 2 1 , 0 ) and ( 2 3 , 0 ) .
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. Architects apply quadratic functions to design arches and suspension bridges, ensuring structural stability and aesthetic appeal. Economists utilize quadratic models to analyze cost and revenue curves, optimizing production and pricing strategies for businesses. These examples highlight the practical significance of quadratic functions in diverse fields, demonstrating their role in solving real-world problems.
