4. The graph below shows part of a parabola of the form y = ax^2 + b. (i) State the value of a. (ii) State the value of b. P is the point (0, c). Find the value of c.
Understand the Problem
The question involves a parabola represented by the equation y = ax^2 + b. It asks for the coefficients a and b based on a point on the graph and wants to find the value of c when x is 0, likely requiring some calculations based on the properties of the parabola.
Answer
$$ c = y_0 - a(x_0)^2 $$
Answer for screen readers
The value of $c$ when $x = 0$ is: $$ c = y_0 - a(x_0)^2 $$
Steps to Solve
- Identify given information
Assume a point on the parabola is given as $(x_0, y_0)$. You need to use this point to find the coefficients $a$ and $b$.
- Set up the equation
Using the point $(x_0, y_0)$ in the parabola equation $y = ax^2 + b$, substitute $x_0$ and $y_0$: $$ y_0 = a(x_0)^2 + b $$
- Solve for b in terms of a
Rearranging the equation gives: $$ b = y_0 - a(x_0)^2 $$
- Find the value of c
The value of $c$ when $x = 0$ is simply: $$ c = b $$
- Substitute the value for b
Now substitute the expression for $b$ obtained from step 3 into the equation for $c$: $$ c = y_0 - a(x_0)^2 $$
- Utilize additional information if any
If there are more points or conditions provided, set up additional equations to solve for $a$ and substitute back to find $c$.
The value of $c$ when $x = 0$ is: $$ c = y_0 - a(x_0)^2 $$
More Information
In a parabola of the form $y = ax^2 + b$, the coefficient $b$ represents the y-intercept, which is the value of $c$ when $x = 0$. The shape of the parabola and the specific coefficients can be determined if additional points on the graph are provided.
Tips
- Forgetting to align the coordinates properly when substituting into the parabola equation.
- Miscalculating the value of $a$ if multiple points are involved; it's crucial to set up all relevant equations correctly before solving.
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