SAT Slope-Intercept Form Practice

It's given that the relationship between $x$ and $y$ is linear. An equation representing a linear relationship can be written in the form $y=mx+b$, where $m$ is the slope and $b$ is the $y$-coordinate of the $y$-intercept of the graph of the relationship in the $xy$-plane. It's given that for every increase in the value of $x$ by 1, the value of $y$ increases by 8.

The slope of a line can be expressed as the change in $y$ over the change in $x$. Thus, the slope, $m$, of the line representing this relationship can be expressed as $\frac{8}{1}$, or 8. Substituting 8 for $m$ in the equation $y=mx+b$ yields $y=8x+b$.

It's also given that when the value of $x$ is 2, the value of $y$ is 18. Substituting 2 for $x$ and 18 for $y$ in the equation $y=8x+b$ yields $18=8\cdot 2+b$, or $18=16+b$. Subtracting 16 from each side of this equation yields $2=b$.

Substituting 2 for $b$ in the equation $y=8x+b$ yields $y=8x+2$. Therefore, the equation $y=8x+2$ represents this relationship.

An equation defining the linear function $f$ can be written in the form $f(x)=mx+b$, where $m$ is the slope and $b$ is the y-intercept of the graph of $y=f(x)$ in the xy-plane. It's given that the graph of $y=f(x)$ has a slope of 2.

Therefore, $m=2$. It's also given that the graph of $y=f(x)$ has a y-intercept at $0,-5$.

Therefore, $b=-5$. Substituting 2 for $m$ and -5 for $b$ in the equation $f(x)=mx+b$ yields $f(x)=2x-5$.

Thus, the equation that defines $f$ is $f(x)=2x-5$.

The equation for the function $f$ in the $xy$-plane can be represented by $f(x)=mx+b$, where $m$ is the slope and $b$ is the $y$-coordinate of the $y$-intercept. Since it's given that the line has a slope of $\frac{3}{4}$, it follows that $m=\frac{3}{4}$ in $f(x)=mx+b$, which yields $y=\frac{3}{4}x+b$.

It's given that $f(0)=3$. This implies that the graph of the function $f$ in the $xy$-plane passes through the point $(0,3)$.

Thus, the $y$-coordinate of the $y$-intercept of the graph is 3, so $b=3$ in $f(x)=\frac{3}{4}x+b$, which yields $f(x)=\frac{3}{4}x+3$. Therefore, the equation $f(x)=\frac{3}{4}x+3$ defines the function $f$.