Graph the equation.
$y=2 x^2-8 x+3$
Answer (1)
Find the vertex of the parabola using x v = − 2 a b and y v = f ( x v ) , resulting in the vertex ( 2 , − 5 ) .
Determine the y-intercept by setting x = 0 , which gives the point ( 0 , 3 ) .
Calculate the x-intercepts by setting y = 0 and solving the quadratic equation, resulting in approximate x-intercepts at ( 0.42 , 0 ) and ( 3.58 , 0 ) .
Plot the vertex, y-intercept, and x-intercepts, and sketch the upward-opening parabola. y = 2 x 2 − 8 x + 3
Explanation
Analyze the equation We are asked to graph the equation y = 2 x 2 − 8 x + 3 . This is a quadratic equation, and its graph is a parabola. Since the coefficient of the x 2 term is positive (2 > 0), the parabola opens upwards.
Find the vertex x-coordinate To graph the parabola, we first find its vertex. The x-coordinate of the vertex is given by the formula x v = − 2 a b , where a = 2 and b = − 8 . Plugging in these values, we get: x v = − 2 ( 2 ) − 8 = 4 8 = 2
Find the vertex y-coordinate Now we find the y-coordinate of the vertex by substituting x v = 2 into the equation: y v = 2 ( 2 ) 2 − 8 ( 2 ) + 3 = 2 ( 4 ) − 16 + 3 = 8 − 16 + 3 = − 5 So, the vertex of the parabola is at the point ( 2 , − 5 ) .
Find the y-intercept Next, we find the y-intercept by setting x = 0 in the equation: y = 2 ( 0 ) 2 − 8 ( 0 ) + 3 = 0 − 0 + 3 = 3 So, the y-intercept is at the point ( 0 , 3 ) .
Find the x-intercepts To find the x-intercepts, we set y = 0 and solve the quadratic equation 2 x 2 − 8 x + 3 = 0 . We use the quadratic formula: x = 2 a − b ± b 2 − 4 a c where a = 2 , b = − 8 , and c = 3 . Plugging in these values, we get: x = 2 ( 2 ) 8 ± ( − 8 ) 2 − 4 ( 2 ) ( 3 ) = 4 8 ± 64 − 24 = 4 8 ± 40 = 4 8 ± 2 10 = 2 ± 2 10 So the x-intercepts are approximately x = 2 + 2 10 ≈ 3.58 and x = 2 − 2 10 ≈ 0.42 .
Sketch the parabola Now we plot the vertex ( 2 , − 5 ) , the y-intercept ( 0 , 3 ) , and the x-intercepts (approximately 0.42 , 0 ) and ( 3.58 , 0 ) on a coordinate plane. Since the parabola opens upwards, we sketch a curve that passes through these points, with the vertex being the minimum point.
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 at approximately ( 0.42 , 0 ) and ( 3.58 , 0 ) .
Examples
Understanding quadratic equations and their graphs is crucial in various fields. For instance, engineers use parabolas to design arches and bridges, ensuring structural stability. In physics, projectile motion follows a parabolic path, allowing us to predict the range and trajectory of objects. Economists also use quadratic functions to model cost and revenue curves, helping businesses optimize their production and pricing strategies. By mastering the art of graphing quadratic equations, students gain valuable insights into real-world phenomena and develop essential problem-solving skills.
