Question

1. allison alquila un instrumento musical. esta gráfica muestra el costo de alquilar el instrumento durante diferentes períodos de tiempo.

¿cómo puede mostrar allison que la pendiente es constante y que representa esa pendiente constante en esta situación?
elige una opción de cada espacio en blanco desplegable para responder la pregunta.
allison puede trazar segmentos desde dos puntos cualesquiera de la recta hasta el eje x y formar triángulos. el resultado de dividir el cambio entre
a. horizontal
b. vertical
a. congruentes
b. semejantes
a. horizontales
b. verticales
el cambio es el mismo para cualquier punto de la recta; por lo tanto, la pendiente de la recta es constante. la pendiente constante muestra que el costo de alquilar un instrumento que allison alquila el instrumento es igual
a. cada mes
b. que el número de meses

Explanation:

Step1: Analyze the first question

The slope of a line is constant. For any two - point on a non - vertical line \(y = mx + b\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). In the context of the cost of renting a musical instrument over time (assuming a linear relationship), the change (slope) is the same for any point on the line.

Step2: Analyze the second question

To form triangles with the \(x\) and \(y\) axes, we can use similar triangles. If we have two right - triangles formed by the line and the axes, the ratios of their corresponding sides are equal. Congruent triangles would require equal side lengths, which is not necessary for the slope (ratio of vertical to horizontal change) calculation. Similar triangles (by AA similarity, since the angles between the line and the axes are the same for all such triangles formed by the line) are used.

Step3: Analyze the third question

The formula for the slope \(m=\frac{\text{vertical change}}{\text{horizontal change}}\). When calculating the slope (change) between two points on the line, we divide the vertical change (difference in the \(y\) - values, which represents the change in cost in this case) by the horizontal change (difference in the \(x\) - values, which represents the change in time).

Step4: Analyze the fourth question

The \(x\) - axis represents time (in months) and the \(y\) - axis represents the cost of renting. To find the change per month (slope), we consider the relationship between the cost (\(y\)) and the number of months (\(x\)).

Answer:

1. B. The change is the same for any point on the line (slope is constant).
2. B. Similar triangles (used to show the constancy of the slope as the ratio of vertical to horizontal changes is the same).
3. B. Vertical (the numerator in the slope formula \(\frac{\text{vertical change}}{\text{horizontal change}}\) when calculating the slope between two points).
4. A. Each month (the slope represents the change in cost per unit change in time, i.e., per month).