Question
Using a certain ramp at the skate park, a skater's height off the ground, in feet, is modeled by the equation y=0.25x
2
−4x+19, where x is the skater's horizontal distance from the jumping platform.
How far from the jumping platform is a skateboarder when they are at their minimum height off the ground?
2
−4x+19, where x is the skater's horizontal distance from the jumping platform.
How far from the jumping platform is a skateboarder when they are at their minimum height off the ground?
Answers
GPT-4o mini
To find the horizontal distance \( x \) from the jumping platform when the skateboarder is at their minimum height off the ground, we need to analyze the quadratic function given by the equation
\[
y = 0.25x^2 - 4x + 19.
\]
A quadratic function of the form \( y = ax^2 + bx + c \) reaches its minimum (or maximum, depending on the sign of \( a \)) at the vertex. The \( x \)-coordinate of the vertex can be calculated using the formula
\[
x_v = -\frac{b}{2a}.
\]
In this case, we have:
- \( a = 0.25 \)
- \( b = -4 \)
Now, substituting these values into the formula:
\[
x_v = -\frac{-4}{2 \times 0.25} = \frac{4}{0.5} = 8.
\]
Thus, the skateboarder is \( \boxed{8} \) feet away from the jumping platform when they are at their minimum height off the ground.
\[
y = 0.25x^2 - 4x + 19.
\]
A quadratic function of the form \( y = ax^2 + bx + c \) reaches its minimum (or maximum, depending on the sign of \( a \)) at the vertex. The \( x \)-coordinate of the vertex can be calculated using the formula
\[
x_v = -\frac{b}{2a}.
\]
In this case, we have:
- \( a = 0.25 \)
- \( b = -4 \)
Now, substituting these values into the formula:
\[
x_v = -\frac{-4}{2 \times 0.25} = \frac{4}{0.5} = 8.
\]
Thus, the skateboarder is \( \boxed{8} \) feet away from the jumping platform when they are at their minimum height off the ground.