Answer
The equations of the parabolas are:
1. \( y = -\frac{5}{12}(x + 3)(x - 4) \)
2. \( y = -\frac{7}{2}(x - 2)(x + 1) \)
3. \( y = \frac{1}{12}(x + 4)(x + 3) \)
4. \( y = \frac{3}{4}(x + 1)(x - 4) \)
5. \( y = -\frac{2}{3}(x + 3)(x + 2) \)
Solution
To find the equation of a parabola using the given \( x \)-intercepts and the \( y \)-intercept, we can use the standard form of the parabola:
\[
y = a(x - x_1)(x - x_2)
\]
where \( (x_1, 0) \) and \( (x_2, 0) \) are the \( x \)-intercepts, and \( (0, y_0) \) is the \( y \)-intercept.
We will follow these steps for each set of points:
1. Identify the \( x \)-intercepts \( x_1 \) and \( x_2 \).
2. Identify the \( y \)-intercept \( y_0 \).
3. Substitute the \( x \)-intercepts into the equation.
4. Use the \( y \)-intercept to solve for \( a \).
5. Write the final equation.
Let's solve each case step by step.
### 1. For \( (-3,0), (4,0) \) and \( (0,5) \)
- \( x_1 = -3 \)
- \( x_2 = 4 \)
- \( y_0 = 5 \)
Substituting into the equation:
\[
y = a(x + 3)(x - 4)
\]
Now, substituting \( x = 0 \) to find \( a \):
\[
5 = a(0 + 3)(0 - 4)
\]
\[
5 = a(3)(-4)
\]
\[
5 = -12a
\]
\[
a = -\frac{5}{12}
\]
Thus, the equation is:
\[
y = -\frac{5}{12}(x + 3)(x - 4)
\]
### 2. For \( (2,0), (-1,0) \) and \( (0,7) \)
- \( x_1 = 2 \)
- \( x_2 = -1 \)
- \( y_0 = 7 \)
Substituting into the equation:
\[
y = a(x - 2)(x + 1)
\]
Now, substituting \( x = 0 \):
\[
7 = a(0 - 2)(0 + 1)
\]
\[
7 = a(-2)(1)
\]
\[
7 = -2a
\]
\[
a = -\frac{7}{2}
\]
Thus, the equation is:
\[
y = -\frac{7}{2}(x - 2)(x + 1)
\]
### 3. For \( (-4,0), (-3,0) \) and \( (0,1) \)
- \( x_1 = -4 \)
- \( x_2 = -3 \)
- \( y_0 = 1 \)
Substituting into the equation:
\[
y = a(x + 4)(x + 3)
\]
Now, substituting \( x = 0 \):
\[
1 = a(0 + 4)(0 + 3)
\]
\[
1 = a(4)(3)
\]
\[
1 = 12a
\]
\[
a = \frac{1}{12}
\]
Thus, the equation is:
\[
y = \frac{1}{12}(x + 4)(x + 3)
\]
### 4. For \( (-1,0), (4,0) \) and \( (0,-3) \)
- \( x_1 = -1 \)
- \( x_2 = 4 \)
- \( y_0 = -3 \)
Substituting into the equation:
\[
y = a(x + 1)(x - 4)
\]
Now, substituting \( x = 0 \):
\[
-3 = a(0 + 1)(0 - 4)
\]
\[
-3 = a(1)(-4)
\]
\[
-3 = -4a
\]
\[
a = \frac{3}{4}
\]
Thus, the equation is:
\[
y = \frac{3}{4}(x + 1)(x - 4)
\]
### 5. For \( (-3,0), (-2,0) \) and \( (0,-4) \)
- \( x_1 = -3 \)
- \( x_2 = -2 \)
- \( y_0 = -4 \)
Substituting into the equation:
\[
y = a(x + 3)(x + 2)
\]
Now, substituting \( x = 0 \):
\[
-4 = a(0 + 3)(0 + 2)
\]
\[
-4 = a(3)(2)
\]
\[
-4 = 6a
\]
\[
a = -\frac{2}{3}
\]
Thus, the equation is:
\[
y = -\frac{2}{3}(x + 3)(x + 2)
\]
### Summary of Equations
1. \( y = -\frac{5}{12}(x + 3)(x - 4) \)
2. \( y = -\frac{7}{2}(x - 2)(x + 1) \)
3. \( y = \frac{1}{12}(x + 4)(x + 3) \)
4. \( y = \frac{3}{4}(x + 1)(x - 4) \)
5. \( y = -\frac{2}{3}(x + 3)(x + 2) \)
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