Determine the value of a for the parabola y = ax^2.
Check the final answer first, then review the worked steps.
Problem
Determine the value of a for the parabola y = ax^2.
Answer
\(a=2\)
Step-by-step solution
- Identify the form of the equation: The problem states that the parabola represents a quadratic function of the form $y = ax^2$. This means the vertex of the parabola is at the origin $(0,0)$.
- Analyze the graph: Observe the provided graph. The parabola opens upwards, which indicates that the coefficient $a$ must be positive.
- Find a point on the parabola: Locate a clear point on the parabola that passes through grid intersections. For example, when $x=1$, the graph appears to pass through $y=2$. Another point is when $x=-1$, $y=2$. Also, when $x=2$, $y=8$ (though this is off the visible grid, it's consistent with the shape). Let's use the point $(1, 2)$.
- Substitute the point into the equation: Substitute the coordinates of the point $(1, 2)$ into the equation $y = ax^2$. So, $2 = a(1)^2$.
- Solve for $a$: Simplify the equation: $2 = a(1)$, which means $a = 2$.
- Verify with another point (optional but recommended): Let's check with another point, for instance, if we consider $x=2$, the corresponding $y$ value appears to be around 8. Substituting $(2, 8)$ into $y = ax^2$: $8 = a(2)^2 \implies 8 = 4a \implies a = 2$. This confirms our value of $a$.