Question

graph a parabola whose x-intercepts are at x = -3 and x = 5 and whose minimum value is y = -4.

Explanation:

Step1: Find vertex x-coordinate

The x-coordinate of the vertex (minimum point) is the midpoint of the x-intercepts:
$$x = \frac{-3 + 5}{2} = 1$$
So the vertex is $(1, -4)$.

Step2: Write intercept form of parabola

The intercept form is $y = a(x - p)(x - q)$, where $p=-3$, $q=5$:
$$y = a(x + 3)(x - 5)$$

Step3: Solve for coefficient $a$

Substitute vertex $(1, -4)$ into the equation:
$$-4 = a(1 + 3)(1 - 5)$$
$$-4 = a(4)(-4)$$
$$-4 = -16a$$
$$a = \frac{-4}{-16} = \frac{1}{4}$$

Step4: Write final equation

Substitute $a=\frac{1}{4}$ back into intercept form:
$$y = \frac{1}{4}(x + 3)(x - 5)$$

Step5: Identify key points for graphing

• X-intercepts: $(-3, 0)$ and $(5, 0)$
• Vertex (minimum): $(1, -4)$
• Additional point (e.g., $x=0$): $y=\frac{1}{4}(3)(-5)=-\frac{15}{4}=-3.75$, so $(0, -3.75)$

Answer:

The equation of the parabola is $\boldsymbol{y = \frac{1}{4}(x + 3)(x - 5)}$. To graph it:

1. Plot the x-intercepts at $(-3, 0)$ and $(5, 0)$
2. Plot the vertex (minimum point) at $(1, -4)$
3. Plot additional points like $(0, -3.75)$ and its symmetric counterpart $(2, -3.75)$
4. Draw a smooth U-shaped curve connecting these points.