Question

1. graph the equation that has a slope of \\(\frac{1}{2}\\) & y-intercept of -3. 37. jan opens a savings account with $200. each month, she deposits $25 into her account and does not withdraw any money from it. a) write an equation in slope-intercept form of the total amount, y, in juanita’s account after x months. b) graph the function. label the axes and scale. c) how much money will juanita have in her savings account after 5 months?

Explanation:

Part (a)

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is $y = mx + b$, where $m$ is the slope (rate of change) and $b$ is the y - intercept (initial value).

Step2: Identify $m$ and $b$

• The initial amount in the savings account ($b$) is $\$200$ because that's the amount when $x = 0$ (at the start, 0 months).
• The rate of change ($m$) is $\$25$ per month because she deposits $\$25$ each month. So $m = 25$ and $b = 200$.

Step3: Write the equation

Substitute $m = 25$ and $b = 200$ into the slope - intercept form $y=mx + b$. We get $y = 25x+200$.

Part (b)

Step1: Label axes

• The x - axis represents the number of months ($x$). We can use a scale where each grid square represents 1 month.
• The y - axis represents the amount of money in the savings account ($y$) in dollars. We can use a scale where each grid square represents, for example, $\$25$ (since the slope is 25, this will make the graphing easier).

Step2: Plot the y - intercept

The y - intercept is at $(0,200)$. So we plot a point at $x = 0$, $y = 200$ on the graph.

Step3: Use the slope to find another point

The slope $m = 25=\frac{25}{1}$. From the point $(0,200)$, we move up 25 units (since the numerator of the slope is 25) and 1 unit to the right (since the denominator of the slope is 1) to get the next point. For example, when $x = 1$, $y=25(1)+200 = 225$, so we plot the point $(1,225)$. We can continue this process to plot more points and then draw a straight line through the points.

Part (c)

Step1: Substitute $x = 5$ into the equation

We have the equation $y = 25x + 200$. Substitute $x = 5$ into the equation.

Step2: Calculate $y$

$y=25(5)+200$. First, calculate $25\times5=125$. Then, $125 + 200=325$.

Answer:

s:
a) The equation is $\boldsymbol{y = 25x + 200}$
b) (Graph: x - axis: months, scale 1 unit = 1 month; y - axis: dollars, scale 1 unit = $\$25$; points plotted at $(0,200),(1,225),(2,250),\cdots$ and a straight line through them)
c) After 5 months, Juanita will have $\boldsymbol{\$325}$ in her savings account.