This interactive graph helps you explore how the quadratic function $y = ax^2 + bx + c$ changes. By adjusting the sliders, you can see how the parameters $a$, $b$, and $c$ affect the shape, position, and intercepts of the parabola.
Use the sliders to change parameters a, b, and c and see the parabola update in real time.
Question 1: When you increase the value of the parameter c using the slider, what happens to the parabola?
Correct Answer: A) The parabola moves up.
Explanation:
In the standard form of a parabola, $y = ax^2 + bx + c$, the parameter c represents the y-intercept. This is the point where the parabola crosses the y-axis, because when $x=0$, the equation simplifies to $y = c$.
Changing the value of c causes a vertical shift of the entire graph. Increasing c shifts the graph up, while decreasing c shifts it down.
Question 2: Given the parabola with the equation $y = 2x^2 - 8x + 5$, calculate the coordinates of its vertex.
Correct Answer: A) (2, -3)
Explanation:
The coordinates of the vertex for a parabola in the form $y = ax^2 + bx + c$ can be found using the vertex formula. The x-coordinate is given by $x = -\frac{b}{2a}$. The y-coordinate is found by substituting this x-value back into the original equation.
In the equation $y = 2x^2 - 8x + 5$, we have $a=2$ and $b=-8$.
$x = -\frac{(-8)}{2(2)} = \frac{8}{4} = 2$
Substitute $x=2$ back into the equation:
$y = 2(2)^2 - 8(2) + 5$
$y = 8 - 16 + 5 = -3$
Therefore, the vertex is at the point (2, -3).