Let's break down and solve the problems shown in the images.

3. Educators studying teens with cell phones

(A) Find the average rate of change from age 15 to age 16.

The formula for the average rate of change is:

$\text{Average Rate of Change} = \frac{\text{Change in Number of Teens}}{\text{Change in Age}}$

From the graph:

• At age 15, the number of teens = 341.
• At age 16, the number of teens = 430.

$\text{Rate of Change} = \frac{430 - 341}{16 - 15} = \frac{89}{1} = 89 \, \text{teens per year}.$

(B) Find the average rate of change from age 17 to age 18.

From the graph:

• At age 17, the number of teens = 590.
• At age 18, the number of teens = 530.

$\text{Rate of Change} = \frac{530 - 590}{18 - 17} = \frac{-60}{1} = -60 \, \text{teens per year}.$

4. Bouncy ball falling down the Grand Canyon

The problem states:

• Initial height = 8000 feet.
• Falling at a rate of 30 feet per second.

The equation for height $h$ as a function of time $t$ can be written as:

$h(t) = 8000 - 30t$

Where:

• $h(t)$ = height at time $t$,
• $t$ = time in seconds,
• $8000$ = initial height,
• $30$ = rate of fall per second.

12. Quadratic Equation: $y = (x + 1)^2 - 4$

(A) Vertex of the parabola

The given equation is in vertex form: $y = a(x - h)^2 + k$.

Here:

• $h = -1$ (opposite sign),
• $k = -4$.

The vertex is $(-1, -4)$.

(B) Convert to standard form $y = ax^2 + bx + c$

Expand $(x + 1)^2$:

$y = (x + 1)^2 - 4 = x^2 + 2x + 1 - 4$

Simplify:

$y = x^2 + 2x - 3$

Thus, the standard form is:

$y = x^2 + 2x - 3.$

(C) y-intercept of the equation

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into $y = x^2 + 2x - 3$:

$y = (0)^2 + 2(0) - 3 = -3.$

The y-intercept is $(0, -3)$.

(D) Intercept form

The intercept form of a quadratic is $y = a(x - m)(x - n)$, where $m$ and $n$ are the roots.

To find the roots, solve $y = x^2 + 2x - 3 = 0$:

Factorize:

$x^2 + 2x - 3 = (x + 3)(x - 1).$

Thus, the intercept form is:

$y = (x + 3)(x - 1).$

(E) x-intercepts

From the factored form $(x + 3)(x - 1) = 0$, set each factor equal to 0:

1. $x + 3 = 0 \Rightarrow x = -3$,
2. $x - 1 = 0 \Rightarrow x = 1$.

The x-intercepts are:

$(-3, 0) \, \text{and} \, (1, 0).$

5 Related Questions:

1. How do you determine the rate of change for a given interval on a graph?
2. What happens to a parabola if the vertex $h$ and $k$ change?
3. How can you identify x- and y-intercepts of a quadratic equation?
4. How do you graph a quadratic equation step by step?
5. What does the slope of a linear function represent in real-world contexts?

Tip: When solving quadratics, the vertex form $y = a(x - h)^2 + k$ is useful for identifying the vertex quickly, while the standard form helps analyze intercepts.