Question

Prove that if a parabola crosses the \(x\) -axis twice, the \(x\) -coordinate of the vertex of the parabola is halfway between the \(x\) -intercepts.

Short answer

Question: Prove that for a parabola that crosses the x-axis twice, the x-coordinate of the vertex is the midpoint of the x-coordinates of the x-intercepts. Answer: To prove this, we followed these steps: 1. Write the general equation of a parabola: $$f(x) = ax^2 + bx + c$$ 2. Find the x-intercepts (roots) of the parabola: $$ax^2 + bx + c = 0$$ 3. Use Vieta's formulas to relate the roots: $$r_1 + r_2 = -\frac{b}{a}$$ and $$r_1 r_2 = \frac{c}{a}$$ 4. Calculate the x-coordinate of the vertex: $$h = -\frac{b}{2a}$$ 5. Show that the x-coordinate of the vertex is the midpoint of the x-intercepts: $$h = \frac{r_1 + r_2}{2}$$ By substituting the expression of \(r_1+r_2\) from Vieta's formulas and simplifying, we proved that the x-coordinate of the vertex is indeed the midpoint of the x-intercepts.

Step-by-step solution

Write the general form of a parabola.

We start by writing the general equation of a parabola in the form $$f(x) = ax^2 + bx + c$$, where \(a \neq 0\).

Find the x-intercepts of the parabola.

To find the x-intercepts, we need to set \(f(x) = 0\) and solve for \(x\). That is, find the roots of the equation. $$ax^2 + bx + c = 0$$ Let \(r_1\) and \(r_2\) be the x-intercepts of the parabola.

Use Vieta's formulas.

According to Vieta's formulas, the sum of the roots and the product of the roots of a quadratic equation can be expressed as: $$r_1 + r_2 = -\frac{b}{a}$$ $$r_1 r_2 = \frac{c}{a}$$

Calculate the x-coordinate of the vertex of the parabola.

The x-coordinate of the vertex \(h\) can be found using the formula: $$h = -\frac{b}{2a}$$

Show that the x-coordinate of the vertex is the midpoint of the x-intercepts.

We want to prove that the x-coordinate of the vertex \(h\) is the midpoint of the x-intercepts \(r_1\) and \(r_2\). The midpoint between two points is calculated as the average of their x-coordinates: $$h = \frac{r_1 + r_2}{2}$$ Substitute the expression of \(r_1+r_2\) from Vieta's formulas: $$h = \frac{-\frac{b}{a}}{2}$$ Simplify the expression to get: $$h = -\frac{b}{2a}$$ Thus, the x-coordinate of the vertex is indeed the midpoint of the x-intercepts, proving the statement.

Key concepts

x-intercepts

The term "x-intercepts" refers to the points where a parabola intersects the x-axis. These are the values of \(x\) for which the function \(f(x) = ax^2 + bx + c\) equals zero. In simpler terms, x-intercepts are the solutions or "roots" of the equation \(ax^2 + bx + c = 0\).

To find these intercepts, you equate the function to zero and solve the resulting quadratic equation:

• Factorize the quadratic expression, if possible.
• Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), if factorization is not feasible.

When a parabola crosses the x-axis twice, it will have two distinct real roots, often labeled as \(r_1\) and \(r_2\). These roots reveal the positions on the x-axis where the parabola "touches down."

Understanding x-intercepts helps us analyze the shape and position of a parabola relative to the x-axis, especially when exploring attributes like symmetry and vertex placement.

vertex of a parabola

The vertex of a parabola is a crucial point that represents either its highest or lowest position, depending on the parabola's orientation. For a parabola represented by a quadratic equation \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula for its x-coordinate:
\[ h = -\frac{b}{2a} \]

Relation with X-Intercepts

An interesting property of the vertex is how its x-coordinate, \(h\), relates to the x-intercepts of the parabola. If a parabola crosses the x-axis twice, the x-coordinate of the vertex is actually the midpoint of the x-intercepts, calculated by the average \(\frac{r_1 + r_2}{2}\). This illustrates a symmetry inherent in parabolas.

When you compare the formula \( h = -\frac{b}{2a} \) to the midpoint formula, it further demonstrates that the vertex's x-coordinate truly lies halfway between \(r_1\) and \(r_2\). This symmetry is why parabolas are such neat shapes in geometry and algebra.

Vieta's formulas

Vieta's formulas offer a surprisingly straightforward way to relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(r_1\) and \(r_2\), Vieta's formulas tell us:

• The sum of the roots \(r_1 + r_2 = -\frac{b}{a}\).
• The product of the roots \(r_1r_2 = \frac{c}{a}\).

Application to Quadratic Equations

These simple expressions are extremely beneficial when exploring properties of parabolas. For instance, they allow us to derive the midpoint relationship that connects the vertex of a parabola to its x-intercepts. Substituting the sum of roots from Vieta's formula into \(\frac{r_1 + r_2}{2}\), simplifies to the vertex formula \(-\frac{b}{2a}\), thus demonstrating the neat balance in quadratic functions.

Understanding Vieta's formulas not only aids in solving roots efficiently but also enhances deeper comprehension of the parabolic symmetry concept, strengthening skills in both algebra and geometry.