Let's break down each statement to understand whether it's true or false.
1. The domain of the function is [tex]\(\left\{ x \;\middle|\; x \geq \frac{1}{4} \right\}\)[/tex]:
The domain of any quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is all real numbers, [tex]\(\mathbb{R}\)[/tex]. Therefore, this statement is false.
2. The range of the function is all real numbers:
The range of the quadratic function [tex]\( f(x) = 2x^2 - x - 6 \)[/tex] is dependent on the vertex since it opens upwards (as the coefficient of [tex]\(x^2\)[/tex] is positive). The parabola has a minimum value at the vertex, and the function can take all values greater than or equal to this minimum value. Therefore, the range is not all real numbers. This statement is false.
3. The vertex of the function is [tex]\(\left( \frac{1}{4}, -6 \frac{1}{8} \right)\)[/tex]:
Using the vertex form formula [tex]\(x = -\frac{b}{2a}\)[/tex], we find the x-coordinate of the vertex to be [tex]\(\frac{1}{4}\)[/tex] and the y-coordinate to be [tex]\(-6.125\)[/tex], which is equivalent to [tex]\(-6 \frac{1}{8}\)[/tex]. Therefore, this statement is true.
4. The function has two [tex]\(x\)[/tex]-intercepts:
The discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex] for the quadratic function [tex]\(2x^2 - x - 6\)[/tex] is positive, indicating that there are two distinct real roots. This means the function intersects the x-axis at two points. Therefore, this statement is true.
5. The function is increasing over the interval [tex]\(\left(-6 \frac{1}{8}, \infty\right)\)[/tex]:
To determine where the function is increasing, we observe the parabola. Since the vertex [tex]\(\left( \frac{1}{4}, -6 \frac{1}{8} \right)\)[/tex] is the minimum point, the function increases for [tex]\( x \ge \frac{1}{4} \)[/tex]. The interval [tex]\(\left(-6 \frac{1}{8}, \infty\right)\)[/tex] mentions the y-coordinate [tex]\(-6 \frac{1}{8}\)[/tex], which is incorrect since intervals should be based on x-values. Therefore, this statement is false.
In conclusion, the two correct statements are:
- The vertex of the function is [tex]\(\left( \frac{1}{4}, -6 \frac{1}{8} \right)\)[/tex].
- The function has two [tex]\(x\)[/tex]-intercepts.
