Let's go through each question:

Question 1

The question asks which statement about the following equations is true.

• Equation A: $y = 3x + 4$
• Equation B: $y = 5x + 2$

To determine the truth of the statements provided, let’s examine each equation’s components:

1. Rate of Change: The rate of change in a linear equation of the form $y = mx + b$ is represented by the slope $m$.

   • For Equation A, the slope is 3.
   • For Equation B, the slope is 5.

   Since 5 (the slope of Equation B) is greater than 3, Equation B has a greater rate of change.

2. Initial Value: The initial value, or y-intercept, is given by the constant $b$.

   • For Equation A, the y-intercept is 4.
   • For Equation B, the y-intercept is 2.

   Since 4 (the intercept of Equation A) is greater than 2, Equation A has a greater initial value.

Correct Answer for Question 1

The correct answer is:

• "Equation A has a greater initial value."

Question 2

The question asks which equations represent a function with a rate of change less than the rate shown on the graph.

To solve this, let's analyze the rate of change (slope) from the graph:

1. Finding the Slope from the Graph:

   • From the graph, you can observe that as $x$ increases by 2 units, $y$ increases by 4 units. This gives a slope (rate of change) of $\frac{4}{2} = 2$.

2. Comparing Rates of Change:

   • The question is asking for equations with a rate of change (slope) less than 2.

If any equations are given in the options, select those with a slope (m) less than 2.

If you need more details, please let me know.

Follow-up Questions:

1. What would change if Equation B had a different y-intercept?
2. How do we determine the slope from any given linear equation?
3. Can initial values be negative, and what would that imply on a graph?
4. Why is the slope important in determining the steepness of a line?
5. How would a zero slope affect the appearance of a line?

Tip:

To quickly identify a line's steepness, compare the absolute values of slopes: larger values mean steeper lines.