Graphing Functions - SAT Math
Card 1 of 30
State the intercepts of $y = x^2 - 4x + 4$.
State the intercepts of $y = x^2 - 4x + 4$.
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$(2, 0)$. Perfect square trinomial $(x-2)^2$ has double root at $x = 2$.
$(2, 0)$. Perfect square trinomial $(x-2)^2$ has double root at $x = 2$.
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What is the vertex of the quadratic function $y = x^2 - 6x + 9$?
What is the vertex of the quadratic function $y = x^2 - 6x + 9$?
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$(3, 0)$. Complete the square: $(x-3)^2$ gives vertex at $(3, 0)$.
$(3, 0)$. Complete the square: $(x-3)^2$ gives vertex at $(3, 0)$.
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What is the standard form of a quadratic function?
What is the standard form of a quadratic function?
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$y = ax^2 + bx + c$. General quadratic form with leading coefficient $a$ and constant term $c$.
$y = ax^2 + bx + c$. General quadratic form with leading coefficient $a$ and constant term $c$.
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What is the domain of the function $f(x) = \frac{1}{x - 2}$?
What is the domain of the function $f(x) = \frac{1}{x - 2}$?
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$x \neq 2$. Denominator cannot equal zero, so $x - 2 \neq 0$.
$x \neq 2$. Denominator cannot equal zero, so $x - 2 \neq 0$.
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State the equation of a horizontal line.
State the equation of a horizontal line.
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$y = c$. Horizontal lines have zero slope and constant y-value.
$y = c$. Horizontal lines have zero slope and constant y-value.
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Find the x-intercepts of $y = x^2 - 9$.
Find the x-intercepts of $y = x^2 - 9$.
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$x = 3, x = -3$. Factor as $(x-3)(x+3) = 0$ to find roots.
$x = 3, x = -3$. Factor as $(x-3)(x+3) = 0$ to find roots.
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What is the range of the function $y = \ln(x)$?
What is the range of the function $y = \ln(x)$?
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All real numbers. Natural logarithm outputs all real values for positive inputs.
All real numbers. Natural logarithm outputs all real values for positive inputs.
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What is the slope of a vertical line?
What is the slope of a vertical line?
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Undefined. Vertical lines have infinite slope, expressed as undefined.
Undefined. Vertical lines have infinite slope, expressed as undefined.
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What is the slope of the line $y = -5x + 2$?
What is the slope of the line $y = -5x + 2$?
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$m = -5$. Coefficient of $x$ in linear form $y = mx + b$.
$m = -5$. Coefficient of $x$ in linear form $y = mx + b$.
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Find the domain of the function $f(x) = \sqrt{x - 4}$.
Find the domain of the function $f(x) = \sqrt{x - 4}$.
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$x \geq 4$. Square root requires non-negative argument: $x - 4 \geq 0$.
$x \geq 4$. Square root requires non-negative argument: $x - 4 \geq 0$.
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What is the result of $f(x) = x^3$ being reflected over the y-axis?
What is the result of $f(x) = x^3$ being reflected over the y-axis?
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$f(x) = -x^3$. Reflection over y-axis changes $x$ to $-x$ in odd functions.
$f(x) = -x^3$. Reflection over y-axis changes $x$ to $-x$ in odd functions.
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Identify the transformation: $f(x) = (x - 4)^2 + 2$.
Identify the transformation: $f(x) = (x - 4)^2 + 2$.
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Right 4, Up 2. $(x - 4)$ shifts right 4, $+2$ shifts up 2.
Right 4, Up 2. $(x - 4)$ shifts right 4, $+2$ shifts up 2.
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State the transformation: $f(x) = x^2$ to $f(x) = 3x^2$.
State the transformation: $f(x) = x^2$ to $f(x) = 3x^2$.
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Vertical stretch by 3. Coefficient greater than 1 stretches graph vertically.
Vertical stretch by 3. Coefficient greater than 1 stretches graph vertically.
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Identify the transformation: $f(x) = x^2$ to $f(x) = (x + 5)^2$.
Identify the transformation: $f(x) = x^2$ to $f(x) = (x + 5)^2$.
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Left 5. $(x + 5)$ represents horizontal shift left by 5 units.
Left 5. $(x + 5)$ represents horizontal shift left by 5 units.
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What is the general form of a linear equation?
What is the general form of a linear equation?
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$y = mx + b$. Standard linear form with slope $m$ and y-intercept $b$.
$y = mx + b$. Standard linear form with slope $m$ and y-intercept $b$.
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State the formula for the slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$.
State the formula for the slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$.
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$m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula between two points.
$m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula between two points.
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State the equation for the asymptote of $y = 2^x - 3$.
State the equation for the asymptote of $y = 2^x - 3$.
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$y = -3$. Exponential functions approach horizontal asymptotes as $x \to -\infty$.
$y = -3$. Exponential functions approach horizontal asymptotes as $x \to -\infty$.
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Identify the vertex form of a quadratic function.
Identify the vertex form of a quadratic function.
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$y = a(x - h)^2 + k$. Shows vertex at $(h, k)$ with horizontal shifts and vertical shifts.
$y = a(x - h)^2 + k$. Shows vertex at $(h, k)$ with horizontal shifts and vertical shifts.
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Identify the equation for a circle centered at the origin.
Identify the equation for a circle centered at the origin.
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$x^2 + y^2 = r^2$. Circle equation with center $(0,0)$ and radius $r$.
$x^2 + y^2 = r^2$. Circle equation with center $(0,0)$ and radius $r$.
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Identify the range of the function $f(x) = -2x^2 + 4$.
Identify the range of the function $f(x) = -2x^2 + 4$.
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$y \leq 4$. Parabola opens downward with maximum value $y = 4$.
$y \leq 4$. Parabola opens downward with maximum value $y = 4$.
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Find the y-intercept of the function $f(x) = 4x^2 - 3x + 5$.
Find the y-intercept of the function $f(x) = 4x^2 - 3x + 5$.
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$y = 5$. Y-intercept is the constant term when $x = 0$.
$y = 5$. Y-intercept is the constant term when $x = 0$.
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State the domain for $f(x) = \frac{1}{x^2 - x - 6}$.
State the domain for $f(x) = \frac{1}{x^2 - x - 6}$.
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$x \neq 3, x \neq -2$. Factor: $(x-3)(x+2) \neq 0$, so exclude both roots.
$x \neq 3, x \neq -2$. Factor: $(x-3)(x+2) \neq 0$, so exclude both roots.
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Find the x-intercept of the line: $y = 3x + 6$.
Find the x-intercept of the line: $y = 3x + 6$.
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$x = -2$. Set $y = 0$ and solve: $0 = 3x + 6$, so $x = -2$.
$x = -2$. Set $y = 0$ and solve: $0 = 3x + 6$, so $x = -2$.
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Find the vertex of $y = 2(x - 3)^2 + 4$.
Find the vertex of $y = 2(x - 3)^2 + 4$.
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$(3, 4)$. Vertex form directly shows vertex coordinates $(h, k)$.
$(3, 4)$. Vertex form directly shows vertex coordinates $(h, k)$.
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Identify the range of $y = 3x^2 - 5$.
Identify the range of $y = 3x^2 - 5$.
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$y \geq -5$. Parabola opens upward with minimum value at vertex $y = -5$.
$y \geq -5$. Parabola opens upward with minimum value at vertex $y = -5$.
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What is the effect of $f(x) = x^2$ becoming $f(x) = -x^2$?
What is the effect of $f(x) = x^2$ becoming $f(x) = -x^2$?
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Reflection over x-axis. Negative coefficient flips parabola upside down.
Reflection over x-axis. Negative coefficient flips parabola upside down.
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Identify the axis of symmetry for $y = ax^2 + bx + c$.
Identify the axis of symmetry for $y = ax^2 + bx + c$.
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$x = -\frac{b}{2a}$. Vertical line through the vertex of any parabola.
$x = -\frac{b}{2a}$. Vertical line through the vertex of any parabola.
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What is the y-intercept of the line: $y = -4x + 7$?
What is the y-intercept of the line: $y = -4x + 7$?
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$y = 7$. Y-intercept occurs when $x = 0$, giving $y = 7$.
$y = 7$. Y-intercept occurs when $x = 0$, giving $y = 7$.
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What is the equation of a parabola opening upwards with vertex $(h, k)$?
What is the equation of a parabola opening upwards with vertex $(h, k)$?
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$y = a(x - h)^2 + k$. Vertex form with $a > 0$ opens upward from vertex $(h, k)$.
$y = a(x - h)^2 + k$. Vertex form with $a > 0$ opens upward from vertex $(h, k)$.
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What is the general form of the exponential function?
What is the general form of the exponential function?
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$y = ab^x$. Base $b$ raised to variable power $x$ with coefficient $a$.
$y = ab^x$. Base $b$ raised to variable power $x$ with coefficient $a$.
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