SAT Math Flashcards: Solving Nonlinear Functions

What this deck covers

This deck focuses on Solving Nonlinear Functions, giving you a quick way to review the definitions, rules, and examples that matter most for SAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the axis of symmetry for $y = x^2 + 6x + 5$.

Answer: $x = -3$. Use formula $x = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$.

Flashcard 2: What is the effect of $a < 0$ in $y = a(x-h)^2 + k$?

Answer: Parabola opens downward. Negative coefficient makes parabola open downward.

Flashcard 3: Identify the roots of $x^2 - 6x + 9 = 0$.

Answer: $x = 3$. Perfect square: $(x-3)^2 = 0$ has repeated root.

Flashcard 4: What is the effect of $a > 0$ in $y = a(x-h)^2 + k$?

Answer: Parabola opens upward. Positive coefficient makes parabola open upward.

Flashcard 5: What does the discriminant $b^2 - 4ac$ indicate about roots?

Answer: Number and type of roots. Determines number of real roots and their nature.

Flashcard 6: Which form is $y = a(x-h)^2 + k$?

Answer: Vertex form. Form that shows vertex $(h,k)$ and transformations.

Flashcard 7: Convert the quadratic $y = 2x^2 + 8x + 6$ to vertex form.

Answer: $y = 2(x+2)^2 - 2$. Complete the square: factor out 2, then add/subtract 4.

Flashcard 8: If $y = x^2 + 2x + 1$, what is the axis of symmetry?

Answer: $x = -1$. Perfect square: $(x+1)^2$, so axis is $x = -1$.

Flashcard 9: Convert $y = x^2 + 6x + 9$ to vertex form.

Answer: $y = (x+3)^2$. Perfect square: $(x+3)^2 = x^2 + 6x + 9$.

Flashcard 10: What is the solution to $4x^2 - 4x + 1 = 0$?

Answer: $x = \frac{1}{2}$. Perfect square: $(2x-1)^2 = 0$ gives $x = \frac{1}{2}$.

Flashcard 11: Find the roots of $x^2 - 5x + 6 = 0$ using factoring.

Answer: $x = 2, x = 3$. Factors as $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

Flashcard 12: State the relationship between roots and factors of a quadratic.

Answer: Roots are solutions; factors are $(x - r_1)(x - r_2)$. If roots are $r_1, r_2$, then factors are $(x-r_1)(x-r_2)$.

Flashcard 13: What is the solution for $4x^2 = 16$?

Answer: $x = 2$ or $x = -2$. Divide by 4: $x^2 = 4$, so $x = \pm 2$.

Flashcard 14: State the quadratic formula.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.

Flashcard 15: What is the discriminant in the quadratic formula?

Answer: $b^2 - 4ac$. Expression under the square root that determines root types.

Flashcard 16: Convert $y = x^2 + 4x + 4$ to vertex form.

Answer: $y = (x+2)^2$. Complete the square: $(x+2)^2 = x^2 + 4x + 4$.

Flashcard 17: What is the solution for $4x^2 = 16$?

Answer: $x = 2$ or $x = -2$. Divide by 4: $x^2 = 4$, so $x = \pm 2$.

Flashcard 18: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. General form with degree 2 polynomial set to zero.

Flashcard 19: Determine the nature of roots for $x^2 + 2x + 5 = 0$.

Answer: Complex. Discriminant $4 - 20 = -16 < 0$ means complex roots.

Flashcard 20: What is the vertex of $y = x^2 - 2x + 1$?

Answer: $(1, 0)$. Perfect square: $(x-1)^2$ has vertex at $(1,0)$.

Flashcard 21: Identify the discriminant of $x^2 + 4x + 4 = 0$.

Answer: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 22: Convert $y = x^2 + 4x + 4$ to vertex form.

Answer: $y = (x+2)^2$. Complete the square: $(x+2)^2 = x^2 + 4x + 4$.

Flashcard 23: Solve for $x$: $x^2 + 9 = 0$.

Answer: $x = 3i$ or $x = -3i$. Take square root: $x^2 = -9$, so $x = \pm 3i$.

Flashcard 24: Which form is $y = a(x-p)(x-q)$?

Answer: Factored form. Shows roots $p$ and $q$ directly as factors.

Flashcard 25: What is a nonlinear function?

Answer: A function that is not a straight line. Graph is not a straight line (degree > 1).

Flashcard 26: Solve for $x$: $x^2 + 6x + 9 = 0$ using the square root method.

Answer: $x = -3$. Perfect square trinomial: $(x+3)^2 = 0$, so $x = -3$.

Flashcard 27: Find the discriminant of $x^2 - 4x + 4 = 0$.

Answer: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 28: Identify the maximum or minimum value of $y = (x-2)^2 + 5$.

Answer: Minimum: $5$. Since $a > 0$, parabola opens up with minimum at vertex.

Flashcard 29: What is the sum of the roots of $ax^2 + bx + c = 0$?

Answer: $-\frac{b}{a}$. Vieta's formula for sum of roots.

Flashcard 30: Identify the discriminant of $x^2 + 4x + 4 = 0$.

Answer: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 31: Determine the nature of roots for $x^2 + 2x + 5 = 0$.

Answer: Complex. Discriminant $4 - 20 = -16 < 0$ means complex roots.

Flashcard 32: Find the vertex of $y = -x^2 + 6x - 9$.

Answer: $(3, 0)$. Perfect square: $-(x-3)^2$ has vertex at $(3,0)$.

Flashcard 33: State the quadratic formula used to solve $ax^2 + bx + c = 0$.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived by completing the square on the general quadratic form.

Flashcard 34: What is the solution to $4x^2 - 4x + 1 = 0$?

Answer: $x = \frac{1}{2}$. Perfect square: $(2x-1)^2 = 0$ gives $x = \frac{1}{2}$.

Flashcard 35: What is the vertex of $y = -2(x+3)^2 + 4$?

Answer: $(-3, 4)$. Vertex form shows $(h,k) = (-3,4)$ directly.

Flashcard 36: What is the standard form equation of a quadratic function?

Answer: $ax^2 + bx + c = 0$. General form where $a \neq 0$ defines any quadratic equation.

Flashcard 37: State the quadratic formula.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Standard formula for solving quadratic equations.

Flashcard 38: Find the roots of $x^2 - 5x + 6 = 0$ using factoring.

Answer: $x = 2, x = 3$. Factors as $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

Flashcard 39: Identify the axis of symmetry for $y = x^2 + 6x + 5$.

Answer: $x = -3$. Use formula $x = -\frac{b}{2a} = -\frac{6}{2(1)} = -3$.

Flashcard 40: State the relationship between roots and factors of a quadratic.

Answer: Roots are solutions; factors are $(x - r_1)(x - r_2)$. If roots are $r_1, r_2$, then factors are $(x-r_1)(x-r_2)$.

Flashcard 41: Find the roots of $x^2 - 5x + 6 = 0$.

Answer: $x = 2$ or $x = 3$. Factor: $(x-2)(x-3) = 0$ gives roots.

Flashcard 42: Which form is $y = a(x-p)(x-q)$?

Answer: Factored form. Shows roots $p$ and $q$ directly as factors.

Flashcard 43: Solve for $x$: $x^2 - 9 = 0$.

Answer: $x = 3$ or $x = -3$. Difference of squares: $(x-3)(x+3) = 0$.

Flashcard 44: What are the roots of $x^2 - 1 = 0$?

Answer: $x = 1$ or $x = -1$. Difference of squares: $(x-1)(x+1) = 0$.

Flashcard 45: Solve for $x$: $x^2 - 4 = 0$.

Answer: $x = 2$ or $x = -2$. Factor as difference of squares: $(x-2)(x+2) = 0$.

Flashcard 46: Solve for $x$: $x^2 + 6x + 9 = 0$ using the square root method.

Answer: $x = -3$. Perfect square trinomial: $(x+3)^2 = 0$, so $x = -3$.

Flashcard 47: Identify the roots of $x^2 - 6x + 9 = 0$.

Answer: $x = 3$. Perfect square: $(x-3)^2 = 0$ has repeated root.

Flashcard 48: Identify the vertex form of a quadratic function.

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and stretch factor $a$ explicitly.

Flashcard 49: What is the product of the roots of $ax^2 + bx + c = 0$?

Answer: $\frac{c}{a}$. Vieta's formula for product of roots.

Flashcard 50: Convert the quadratic $y = 2x^2 + 8x + 6$ to vertex form.

Answer: $y = 2(x+2)^2 - 2$. Complete the square: factor out 2, then add/subtract 4.

Flashcard 51: Which method can solve $x^2 + 4x + 4 = 0$ besides factoring?

Answer: Completing the square. Transforms equation to perfect square trinomial form.

Flashcard 52: Find the value of $x$ in $x^2 + 4x + 4 = 0$.

Answer: $x = -2$. Perfect square: $(x+2)^2 = 0$ gives $x = -2$.

Flashcard 53: What does a positive discriminant indicate about a quadratic's roots?

Answer: Two distinct real roots. When $b^2 - 4ac > 0$, the parabola crosses the x-axis twice.

Flashcard 54: What is the vertex form of a quadratic function?

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shifts/stretches.

Flashcard 55: What is the discriminant in the quadratic formula?

Answer: $b^2 - 4ac$. Expression under the square root that determines root types.

Flashcard 56: What is the standard form of a quadratic equation?

Answer: $ax^2 + bx + c = 0$. General form with degree 2 polynomial set to zero.

Flashcard 57: What is a nonlinear function?

Answer: A function that is not a straight line. Graph is not a straight line (degree > 1).

Flashcard 58: For $y = 3(x-4)^2 - 7$, what is the vertex?

Answer: $(4, -7)$. Vertex form shows $(h,k) = (4,-7)$ directly.

Flashcard 59: Convert $y = x^2 + 6x + 9$ to vertex form.

Answer: $y = (x+3)^2$. Perfect square: $(x+3)^2 = x^2 + 6x + 9$.

Flashcard 60: Find the vertex of $y = -x^2 + 6x - 9$.

Answer: $(3, 0)$. Perfect square: $-(x-3)^2$ has vertex at $(3,0)$.

Flashcard 61: Solve for $x$: $x^2 - 4 = 0$.

Answer: $x = 2$ or $x = -2$. Factor as difference of squares: $(x-2)(x+2) = 0$.

Flashcard 62: What is the degree of a quadratic function?

Answer:

1. Highest power of variable is 2.

Flashcard 63: What type of solutions does a quadratic have if the discriminant is negative?

Answer: Complex solutions. Negative discriminant means no real roots.

Flashcard 64: What does a positive discriminant indicate about a quadratic's roots?

Answer: Two distinct real roots. When $b^2 - 4ac > 0$, the parabola crosses the x-axis twice.

Flashcard 65: What is the vertex of $y = -2(x+3)^2 + 4$?

Answer: $(-3, 4)$. Vertex form shows $(h,k) = (-3,4)$ directly.

Flashcard 66: What is the effect of $a < 0$ in $y = a(x-h)^2 + k$?

Answer: Parabola opens downward. Negative coefficient makes parabola open downward.

Flashcard 67: Find the vertex of $y = 2(x - 3)^2 + 4$.

Answer: Vertex: $(3, 4)$. In vertex form, $(h,k)$ gives the vertex coordinates directly.

Flashcard 68: For $y = (x-1)^2 + 3$, identify the vertex.

Answer: $(1, 3)$. Vertex form shows $(h,k) = (1,3)$ directly.

Flashcard 69: If $y = x^2 + 2x + 1$, what is the axis of symmetry?

Answer: $x = -1$. Perfect square: $(x+1)^2$, so axis is $x = -1$.

Flashcard 70: Find the roots of $x^2 - 5x + 6 = 0$.

Answer: $x = 2$ or $x = 3$. Factor: $(x-2)(x-3) = 0$ gives roots.

Flashcard 71: Convert $y = x^2 - 4x + 6$ to vertex form.

Answer: $y = (x-2)^2 + 2$. Complete the square: $(x-2)^2 = x^2 - 4x + 4$.

Flashcard 72: What are the roots of $x^2 - 1 = 0$?

Answer: $x = 1$ or $x = -1$. Difference of squares: $(x-1)(x+1) = 0$.

Flashcard 73: Find the vertex of $y = 2(x - 3)^2 + 4$.

Answer: Vertex: $(3, 4)$. In vertex form, $(h,k)$ gives the vertex coordinates directly.

Flashcard 74: What type of solutions does a quadratic have if the discriminant is negative?

Answer: Complex solutions. Negative discriminant means no real roots.

Flashcard 75: What is the product of the roots of $ax^2 + bx + c = 0$?

Answer: $\frac{c}{a}$. Vieta's formula for product of roots.

Flashcard 76: Identify the maximum or minimum value of $y = (x-2)^2 + 5$.

Answer: Minimum: $5$. Since $a > 0$, parabola opens up with minimum at vertex.

Flashcard 77: Which form is $y = a(x-h)^2 + k$?

Answer: Vertex form. Form that shows vertex $(h,k)$ and transformations.

Flashcard 78: State the quadratic formula used to solve $ax^2 + bx + c = 0$.

Answer: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived by completing the square on the general quadratic form.

Flashcard 79: Find the value of $x$ in $x^2 + 4x + 4 = 0$.

Answer: $x = -2$. Perfect square: $(x+2)^2 = 0$ gives $x = -2$.

Flashcard 80: Find the axis of symmetry for $y = 2x^2 + 4x + 1$.

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

Flashcard 81: Solve for $x$: $x^2 + 9 = 0$.

Answer: $x = 3i$ or $x = -3i$. Take square root: $x^2 = -9$, so $x = \pm 3i$.

Flashcard 82: For $y = (x-1)^2 + 3$, identify the vertex.

Answer: $(1, 3)$. Vertex form shows $(h,k) = (1,3)$ directly.

Flashcard 83: For $y = 3(x-4)^2 - 7$, what is the vertex?

Answer: $(4, -7)$. Vertex form shows $(h,k) = (4,-7)$ directly.

Flashcard 84: What is the vertex form of a quadratic function?

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shifts/stretches.

Flashcard 85: What is the degree of a quadratic function?

Answer:

1. Highest power of variable is 2.

Flashcard 86: What does the discriminant $b^2 - 4ac$ indicate about roots?

Answer: Number and type of roots. Determines number of real roots and their nature.

Flashcard 87: What is the effect of $a > 0$ in $y = a(x-h)^2 + k$?

Answer: Parabola opens upward. Positive coefficient makes parabola open upward.

Flashcard 88: What is the vertex of $y = x^2 - 2x + 1$?

Answer: $(1, 0)$. Perfect square: $(x-1)^2$ has vertex at $(1,0)$.

Flashcard 89: Identify the vertex form of a quadratic function.

Answer: $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and stretch factor $a$ explicitly.

Flashcard 90: Find the discriminant of $x^2 - 4x + 4 = 0$.

Answer: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

Flashcard 91: For $y = x^2 - 4x + 4$, determine the axis of symmetry.

Answer: $x = 2$. Axis of symmetry is $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

Flashcard 92: What is the sum of the roots of $ax^2 + bx + c = 0$?

Answer: $-\frac{b}{a}$. Vieta's formula for sum of roots.