Which of the following graphs represents the solution set of the quadratic equation?

Question

y≤2x2+2x+1

Answer 1

To determine which graph represents the solution set of the quadratic inequality $y\le 2x^2+2x+1$, we need to analyze the quadratic $2x^2+2x+1$.

1. Identify the Quadratic Function: The quadratic function is given by $f(x)=2x^2+2x+1$.

2. Finding the Vertex: The vertex of a quadratic function in the form $a{x}^2+bx+c$ can be found using the formula: $$x=-\frac{b}{2a}$$ Here, $a=2$ and $b=2$: $$x=-\frac{2}{2\cdot 2}=-\frac{1}{2}$$ Now, substitute $x=-\frac{1}{2}$ back into the function to find the y-coordinate of the vertex: $$f(-\frac{1}{2})=2{(-\frac{1}{2})}^2+2(-\frac{1}{2})+1=2\left(\frac{1}{4}\right)-1+1=\frac{1}{2}$$ Therefore, the vertex of the parabola is at $(-\frac{1}{2},\frac{1}{2})$.

3. Direction of the Parabola: Since the coefficient of $x^2$ (which is 2) is positive, the parabola opens upwards.

4. Determine the Solution Set: The inequality $y\le 2x^2+2x+1$ represents all points on or below the parabola. This means we need to shade the region below the curve.

5. Y-intercept: To find the y-intercept, set $x=0$: $$f(0)=2(0{)}^2+2(0)+1=1$$ Therefore, the y-intercept is at point $(0,1)$.

6. X-intercepts: To find the x-intercepts, set $y=0$: $$0=2x^2+2x+1$$ The discriminant of the quadratic $2x^2+2x+1$ is: $$b^2-4ac=2^2-4(2)(1)=4-8=-4$$ Since the discriminant is negative, the quadratic has no real roots, indicating that the parabola does not touch the x-axis.

Now that we have analyzed the quadratic function and its behavior, you can look for a graph that displays an upward-opening parabola with its vertex at $(-\frac{1}{2},\frac{1}{2})$, does not intersect the x-axis, and has shading below the curve representing the area where $y\le 2x^2+2x+1$.