Function Model Construction and Application - AP Precalculus

$y = 10$. Evaluate $f(0)$ to find where graph crosses y-axis.

Quadratic function. Basic parabola opening upward.

Reflection over the x-axis. Negative coefficient flips parabola upside down.

$x = 2, x = -2$. Set $f(x) = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

Slope $m = \frac{2}{3}$. Rearrange to $y = \frac{2}{3}x - 2$ form.

All non-negative real numbers. Cannot take square root of negative numbers in real domain.

All positive real numbers. Exponential function always produces positive outputs.

Horizontal shift left by 2 units. Adding inside parentheses shifts graph left.

Vertical shift up by 5 units. Adding constant moves entire graph upward.

Vertex at $(1, 3)$. Vertex form directly shows vertex coordinates $(h,k)$.

Vertical shift down by 1 unit. Subtracting constant moves entire graph downward.

From -1 to 1 inclusive. Cosine oscillates between minimum -1 and maximum 1.

$y = 1$. Evaluate $f(0) = 3(0)^2 + 2(0) + 1 = 1$.

$x = 2$. Use formula $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

All positive real numbers. Logarithm undefined for zero and negative inputs.

$f^{-1}(x) = \frac{x-3}{2}$. Switch variables and solve: $x = 2y + 3$ gives $y = \frac{x-3}{2}$.

$f(x) = a(x-h)^2 + k$. Shows vertex at $(h,k)$ with $a$ controlling width and direction.

$f(x) = a(x-h)^2 + k$. Shows vertex at $(h,k)$ with $a$ controlling width and direction.

$f^{-1}(x) = \frac{1}{x}$. Function is its own inverse (self-inverse).

All positive real numbers. Exponential function always produces positive outputs.

$(x-h)^2 + (y-k)^2 = r^2$. Center at $(h,k)$ with radius $r$.

All real numbers. Linear functions have unlimited output values.

Horizontal shift right by 3 units. Subtracting inside parentheses shifts graph right.

$f^{-1}(x) = \frac{1}{x}$. Function is its own inverse (self-inverse).

All real numbers except $x = 2$. Denominator cannot equal zero, so $x \neq 2$.

$y = 4$. Horizontal line has constant y-value for all x.

Rational function. Ratio of polynomials defines rational functions.

$\frac{\text{Period}}{2}$. Coefficient of $x$ doubles frequency, halving period to $\pi$.

Amplitude is 3. Coefficient determines maximum displacement from center.