Functions: Linear and Nonlinear Functions (CCSS.8.F.3)
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8th Grade Math › Functions: Linear and Nonlinear Functions (CCSS.8.F.3)
Which equation represents a linear function?
$y=x^2+3$
$y=3x-5$
$y=2^x$
$y=\frac{4}{x}$
Explanation
Linear functions have the form $y=mx+b$ and a constant rate of change (straight-line graph). $y=3x-5$ fits this form. The others are quadratic, exponential, and reciprocal, which have curved graphs and changing rates of change.
Which equation is nonlinear?
$y=-4x+1$
$y=0.5x-7$
$y=3$
$y=\sqrt{x}$
Explanation
$y=\sqrt{x}$ is nonlinear because its graph is curved and the rate of change is not constant. The other choices are of the form $y=mx+b$ (including a constant function), so they are linear.
Which equation defines a linear function with a constant rate of change?
$y=-\frac{2}{3}x+4$
$y=x^2-1$
$y=5\cdot 2^x$
$y=x^3$
Explanation
Only $y=-\frac{2}{3}x+4$ has the form $y=mx+b$, so its graph is a straight line with constant slope. The other choices (quadratic, exponential, cubic) are nonlinear and have changing rates of change.
Which equation does NOT represent a linear function?
$y=-7x$
$y=\frac{1}{2}x+9$
$y=\frac{1}{x}+2$
$y=-5$
Explanation
A linear function has a straight-line graph and the form $y=mx+b$. Choices A, B, and D are linear (including the constant function $y=-5$). $y=\frac{1}{x}+2$ is nonlinear; its graph is curved and the rate of change varies.
Which equation does NOT define a linear function?
$y=3x$
$y=-x+9$
$y=\tfrac{1}{2}x-4$
$y=x^2+3x$
Explanation
Linear functions have $x$ only to the first power and graph as straight lines. $y=3x$, $y=-x+9$, and $y=\tfrac{1}{2}x-4$ are linear (constant slope). $y=x^2+3x$ is quadratic, so its rate of change varies and its graph is a parabola (nonlinear).
Which equation represents a linear function with a constant rate of change?
$y=|x|$
$y=4x-1$
$xy=8$
$y=5^x$
Explanation
Linear functions have the form $y=mx+b$ and a constant slope. $y=4x-1$ fits that form. $y=|x|$ is V-shaped (piecewise, not a single straight line), $xy=8$ gives $y=\tfrac{8}{x}$ (inverse variation), and $y=5^x$ is exponential; all are nonlinear with changing rates of change.
Which equation is NOT linear?
$y = 0.25x - 5$
$y = (x-1)^2 + 2$
$y = -x$
$y = 9$
Explanation
Linear functions look like $y=mx+b$ and have a constant rate of change: $0.25x-5$, $-x$, and $9$ (a horizontal line) are linear. $(x-1)^2+2$ is quadratic, so its graph is curved and the rate of change is not constant.
Which function is linear?
$f(x)=|x|-3$
$g(x)=5x$
$h(x)=x^2-2x$
$p(x)=3^x$
Explanation
$g(x)=5x$ is linear because it matches $y=mx+b$ with $m=5$ and $b=0$, giving a constant rate of change and a straight-line graph. $|x|-3$ (V-shaped), $x^2-2x$ (quadratic), and $3^x$ (exponential) are nonlinear and produce curved graphs with changing rates.