Graph the equation.
$y=2 x^2-8 x+3$

Find the vertex of the parabola using x = − b / ( 2 a ) and substitute to find the y-coordinate.
Determine the y-intercept by setting x = 0 in the equation.
Calculate the x-intercepts by setting y = 0 and using the quadratic formula.
Plot the vertex, y-intercept, and x-intercepts to graph the parabola: y = 2 x 2 − 8 x + 3 ​ .

Explanation

Analyze the problem We are asked to graph the quadratic equation y = 2 x 2 − 8 x + 3 . This equation represents a parabola. Since the coefficient of the x 2 term is positive (2 > 0), the parabola opens upwards. To graph this parabola, we need to find its vertex, x-intercepts, and y-intercept.

Find the vertex The vertex of a parabola in the form y = a x 2 + b x + c can be found using the formula x = − b / ( 2 a ) . In our case, a = 2 and b = − 8 , so the x-coordinate of the vertex is x = − ( − 8 ) / ( 2 ∗ 2 ) = 8/4 = 2 . To find the y-coordinate of the vertex, we substitute x = 2 into the equation: y = 2 ( 2 ) 2 − 8 ( 2 ) + 3 = 2 ( 4 ) − 16 + 3 = 8 − 16 + 3 = − 5 . Therefore, the vertex of the parabola is ( 2 , − 5 ) .

Find the y-intercept The y-intercept is the point where the parabola intersects the y-axis, which occurs when x = 0 . Substituting x = 0 into the equation, we get y = 2 ( 0 ) 2 − 8 ( 0 ) + 3 = 3 . So, the y-intercept is ( 0 , 3 ) .

Find the x-intercepts The x-intercepts are the points where the parabola intersects the x-axis, which occurs when y = 0 . So we need to solve the quadratic equation 2 x 2 − 8 x + 3 = 0 . We can use the quadratic formula to find the x-intercepts: x = ( − b ± b 2 − 4 a c ​ ) / ( 2 a ) . In our case, a = 2 , b = − 8 , and c = 3 . So, x = ( 8 ± ( − 8 ) 2 − 4 ( 2 ) ( 3 ) ​ ) / ( 2 ∗ 2 ) = ( 8 ± 64 − 24 ​ ) /4 = ( 8 ± 40 ​ ) /4 = ( 8 ± 2 10 ​ ) /4 = 2 ± 2 10 ​ ​ . Thus, the x-intercepts are approximately x = 2 + 2 10 ​ ​ ≈ 3.58 and x = 2 − 2 10 ​ ​ ≈ 0.42 . The x-intercepts are approximately ( 3.58 , 0 ) and ( 0.42 , 0 ) .

Plot the points and draw the parabola Now we have the vertex ( 2 , − 5 ) , the y-intercept ( 0 , 3 ) , and the x-intercepts approximately ( 3.58 , 0 ) and ( 0.42 , 0 ) . We can plot these points on a coordinate plane and draw a smooth curve through the points to graph the parabola.

Final Answer The graph of the equation y = 2 x 2 − 8 x + 3 is a parabola that opens upwards, with vertex at ( 2 , − 5 ) , y-intercept at ( 0 , 3 ) , and x-intercepts approximately at ( 3.58 , 0 ) and ( 0.42 , 0 ) .


Examples
Understanding quadratic equations and their graphs is essential in many fields. For example, engineers use parabolas to design arches and bridges, ensuring structural stability. In physics, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. By knowing the equation of the parabola, one can predict the range and maximum height of the projectile. Similarly, in economics, quadratic functions can model cost and revenue curves, helping businesses optimize their production and pricing strategies.

Answered by GinnyAnswer | 2025-07-08