Graph the equation.
$y=-\frac{1}{8} x^2+x-4$

Find the vertex of the parabola using the formula x v ​ = − 2 a b ​ and substituting into the equation to find y v ​ , resulting in the vertex ( 4 , − 2 ) .
Determine the y-intercept by setting x = 0 in the equation, which gives y = − 4 , so the y-intercept is ( 0 , − 4 ) .
Attempt to find the x-intercepts by setting y = 0 and using the quadratic formula, but find that the discriminant is negative, indicating no real x-intercepts.
Utilize the symmetry of the parabola to find a point symmetric to the y-intercept across the vertex, resulting in the point ( 8 , − 4 ) .
Vertex: ( 4 , − 2 ) , y-intercept: ( 0 , − 4 ) , no x-intercepts ​

Explanation

Analyze the problem The problem asks us to graph the quadratic equation y = − 8 1 ​ x 2 + x − 4 . This is a parabola that opens downward since the coefficient of the x 2 term is negative. To graph it, we need to find the vertex, the y-intercept, and if they exist, the x-intercepts.

Find the x-coordinate of the vertex The vertex of a parabola given by the equation y = a x 2 + b x + c is at the point ( x v ​ , y v ​ ) , where x v ​ = − 2 a b ​ . In our case, a = − 8 1 ​ and b = 1 . Therefore, x v ​ = − 2 ( − 8 1 ​ ) 1 ​ = − − 4 1 ​ 1 ​ = 4.

Find the y-coordinate of the vertex To find the y-coordinate of the vertex, we substitute x v ​ = 4 into the equation: y v ​ = − 8 1 ​ ( 4 ) 2 + 4 − 4 = − 8 1 ​ ( 16 ) = − 2. So, the vertex is at ( 4 , − 2 ) .

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 = − 8 1 ​ ( 0 ) 2 + 0 − 4 = − 4. Thus, the y-intercept is at ( 0 , − 4 ) .

Find the x-intercepts To find the x-intercepts, we set y = 0 and solve for x : − 8 1 ​ x 2 + x − 4 = 0. Multiplying by -8, we get: x 2 − 8 x + 32 = 0. Using the quadratic formula, x = 2 a − b ± b 2 − 4 a c ​ ​ , where a = 1 , b = − 8 , and c = 32 , we have: x = 2 ( 1 ) 8 ± ( − 8 ) 2 − 4 ( 1 ) ( 32 ) ​ ​ = 2 8 ± 64 − 128 ​ ​ = 2 8 ± − 64 ​ ​ . Since the discriminant is negative, there are no real x-intercepts.

Find a symmetric point Now we have the vertex ( 4 , − 2 ) and the y-intercept ( 0 , − 4 ) . Since the parabola is symmetric about the vertical line through the vertex, we can find another point on the parabola by reflecting the y-intercept across the line x = 4 . The x-coordinate of the reflected point is 4 + ( 4 − 0 ) = 8 . The reflected point is ( 8 , − 4 ) .

Sketch the parabola We can now plot the vertex ( 4 , − 2 ) , the y-intercept ( 0 , − 4 ) , and the point ( 8 , − 4 ) to sketch the parabola.

Final Answer The vertex of the parabola is ( 4 , − 2 ) , the y-intercept is ( 0 , − 4 ) , and there are no x-intercepts. The parabola opens downwards.


Examples
Understanding parabolas is crucial in many real-world applications. For instance, engineers use parabolic shapes in designing suspension bridges and arches because of their excellent structural properties. The cables in a suspension bridge often form a parabola, distributing the load evenly. Similarly, the path of a projectile, like a ball thrown in the air, approximately follows a parabolic trajectory, which is essential for calculating range and height in sports and military applications. Reflectors in flashlights and satellite dishes also use parabolic shapes to focus light or radio waves efficiently.

Answered by GinnyAnswer | 2025-07-08