We will start by sketching the related quadratic function. To do so, we first need to identify the values of a, b, and c.
y= 4x^2+ 12x+ 10
We see that a= 4, b= 12, and c= 10. Let's substitute these values into the Quadratic Formula to find the roots of 4x^2 +12x+10 =0.
We have found a negative discriminant, - 16. Therefore, the related quadratic function has no real solutions. This means the graph of the quadratic function does not intercept the x-axis. Consider now the given inequality.
4x^2 +12x+10 ≤ 0
The solution of the quadratic inequality consists of x-values for which the graph of the related quadratic function lies on and below the x-axis. The graph opens upward, since a= 4 is greater than zero.
We see that the graph lies above the x-axis for all x-values, meaning that no x-value satisfies the inequality.
{ } ⇔ ∅
Showing Our Work
Drawing the parabola using a point on the curve and vertex
We need at least three points to draw the graph of the related quadratic function. To do this, we will first find the vertex, (- b2a,f(- b2a)). Then, we will find a point on the curve and reflect it across the axis of symmetry. Let's start by calculating the x-coordinate of the vertex. To do so, we will substitute a= 4 and b= 12 into - b2a.
The vertex is (- 1.5,1). Note that the formula for the x-coordinate of the vertex is the same as the formula for the axis of symmetry, which is x=- 1.5. Now, we will choose an x-value and substitute it for x in the function. Let's try x=- 2.5.
The point (- 2.5,5) lies on the curve. Since the axis of symmetry divides the graph into two mirror images, there exists another point on the opposite side of the axis of symmetry with the same y-value.
Point
Reflection Across x=- 1.5
(- 2.5, 5)
(- 0.5, 5)
The vertex, along with the points, is helpful to graph a parabola.
