Learn to identify and understand the key components of the standard form (y = ax^2 + bx + c) and vertex form (y = a(x-h)^2 + k) of a parabola's equation. This stack is foundational for analyzing quadratic functions on the SAT.
What is the standard form of a quadratic equation that represents a parabola?
The standard form is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
In the standard form \(y = ax^2 + bx + c\), what does the sign of the coefficient \(a\) tell you about the parabola?
If \(a > 0\), the parabola opens upwards (like a smile). If \(a < 0\), the parabola opens downwards (like a frown).
In the standard form \(y = ax^2 + bx + c\), what feature of the parabola does the constant \(c\) directly represent?
The constant \(c\) represents the y-intercept of the parabola. The y-intercept is the point where the graph crosses the y-axis, which occurs at \((0, c)\).
What is the vertex form of a quadratic equation that represents a parabola?
The vertex form is \(y = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants and \(a \neq 0\).
In the vertex form \(y = a(x-h)^2 + k\), what coordinates represent the vertex of the parabola?
The vertex is located at the point \((h, k)\). Be careful with the sign of \(h\); the form is \((x-h)\).
A parabola's equation is \(y = 3(x-4)^2 + 5\). What are the coordinates of its vertex?
The vertex is at \((h, k)\). Here, \(h = 4\) and \(k = 5\). So the vertex is \((4, 5)\).
A parabola has the equation \(y = -2(x+1)^2 - 6\). What is its vertex?
The vertex form is \(y = a(x-h)^2 + k\). We can rewrite the equation as \(y = -2(x - (-1))^2 - 6\). Therefore, \(h = -1\) and \(k = -6\). The vertex is \((-1, -6)\).
What is the axis of symmetry of a parabola, and how do you find it from the vertex form \(y = a(x-h)^2 + k\)?
The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror images. Its equation is \(x = h\).
For a parabola given by \(y = 5(x-2)^2 + 8\), what is the equation of its axis of symmetry?
The vertex is at \((2, 8)\). The axis of symmetry is the vertical line passing through the vertex, so its equation is \(x = 2\).
In the vertex form \(y = a(x-h)^2 + k\), what does the value of \(k\) represent?
The value \(k\) is the y-coordinate of the vertex. It represents the minimum value of the function if the parabola opens up (\(a > 0\)) or the maximum value if it opens down (\(a < 0\)).
Does the parabola \(y = -4(x+3)^2 + 10\) have a minimum or a maximum value? What is that value?
Since \(a = -4\) (which is less than \(0\)), the parabola opens downwards and has a maximum value. The maximum value is the y-coordinate of the vertex, which is \(k = 10\).
Does the parabola \(y = 0.5(x-7)^2 - 1\) have a minimum or a maximum value? What is that value?
Since \(a = 0.5\) (which is greater than \(0\)), the parabola opens upwards and has a minimum value. The minimum value is the y-coordinate of the vertex, which is \(k = -1\).
