Question

There are three different relationships between $x$ and $y.$
Function I represents a straight line consisting of a set of ordered pairs such as $(3,4),(4,5),$ and $(7,8).$
Function II:
$\begin{array}{cc}\text{Input (x)}& \text{Output (y)}\\ 0& 6\\ 12& 30\\ 18& 42\end{array}$
Function III: It represents a straight line having a slope greater than the slope of each function I and II, but the y-intercept is greater than that of function I and less than that of function II.
The equation of function III can be _______ .

• A. $y=4x+4$
• B. $y=2x+5$
• C. $y=3x+7$
• D. $y=1.5x+2$

Answer 1

The correct option is A $y=4x+4$

Detailed step-by-step solution:
Finding the equation of function I:
Consider two points (3, 4) and (4, 5):
Slope, $m=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in x}}$
$=\frac{5-4}{4-3}$
$=1$
At $x=3,y=4.$
Let the equation be $y=mx+b.$
For the equation $y=mx+b$
$\Rightarrow m=$ Slope
$\Rightarrow m=1$
$y=mx+b$
$\Rightarrow 4=1\left(3\right)+b$ (substituting $m=1,x=3,$ and $y=4)$
$\Rightarrow b=1$
For the equation: $y=mx+b$
y-intercept $=b=1$
Finding the slope and the y-intercept of function II:
y-intercept is the value of $y$ when $x=0.$
At $x=0,y=6.$
$\Rightarrow$ y-intercept $=6$
Consider two points $(0,6)$ and $(12,30):$
Slope, $m=\frac{\text{Rise}}{\text{Run}}$
$=\frac{\text{Change in y}}{\text{Change in x}}$
$=\frac{30-6}{12-0}$
$=\frac{24}{12}$
$=2$
Finding the slope and the y-intercept of function III:
Slope of function III $>$ Slope of function I
Slope of the function III $>$ Slope of function II
$\Rightarrow$ Slope (m) $>$ 1
Slope $\left(m\right)>2$
$\Rightarrow$ Slope $\left(m\right)>2$
y-intercept of function I $<$ y-intercept of function III $<$ y-intercept of function II
$\Rightarrow 1<$ y-intercept $<6$
Let’s look at the options:
A. $y=1.5x+2:$
$\Rightarrow$ Slope $\left(m\right)=1.5$
$\Rightarrow 1.5<2$
$\Rightarrow$ Slope $\left(m\right)<2$
This equation is not satisfying the following condition: Slope $\left(m\right)>2$
Hence, this cannot be the equation of function III.
B. $y=2x+5:$
$\Rightarrow$ Slope $\left(m\right)=2$
This equation is not satisfying the following condition: Slope $\left(m\right)>2$
Hence, this cannot be the equation of function III.
$y=4x+4:$
$\Rightarrow$ Slope $\left(m\right)=4$
$\Rightarrow 4>2$
$\Rightarrow$ Slope $\left(m\right)>2$
y-intercept $=4$
$\Rightarrow 1<4<6$
$\Rightarrow 1<$ y-intercept $<6$
This equation is satisfying the following conditions:
Slope $\left(m\right)>2$
$1<$ y-intercept $<6$
Hence, this can be the equation of function III.
$y=3x+7:$
Slope $\left(m\right)=3$
$\Rightarrow 3>2$
$\Rightarrow$ Slope $\left(m\right)>2$
y-intercept $=7$
$\Rightarrow 7>6$
$\Rightarrow$ y-intercept $>6$
This equation is not satisfying the following condition: $1<$ y-intercept $<6.$
Hence, this cannot be the equation of function III.
$\Rightarrow$ Option C is correct.