This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For g(x) = x⁴ - 2x², we need to check for symmetry. A function is even if g(-x) = g(x) for all x (symmetric about y-axis), and odd if g(-x) = -g(x) for all x (symmetric about origin). Let's check: g(-x) = (-x)⁴ - 2(-x)² = x⁴ - 2x² = g(x). Since g(-x) = g(x), the function is even! Choice A correctly identifies the function as 'Even (symmetric about the y-axis)' because replacing x with -x gives us the exact same function, which means the graph is a mirror image across the y-axis. Choice B would require g(-x) = -g(x), but we found g(-x) = g(x) instead—the function is definitely not odd! For symmetry checking: Even functions have only even powers of x (like x², x⁴) or constants. Odd functions have only odd powers (like x, x³). Mixed functions are usually neither. This function has only even powers (x⁴ and x²), so it's even!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. A maximum is the highest point on a graph (or on some portion of it), giving the largest y-value, while a minimum is the lowest point, giving the smallest y-value. For a parabola that opens up, the vertex is the minimum; if it opens down, the vertex is the maximum. In real-world problems, these tell you the best or worst outcome! For this downward-opening parabola H(x) = $-(x-2)^2$ + 9, the vertex at (2,9) is the maximum, meaning the highest point on the roller coaster track is 9 meters at x=2 meters horizontally. Choice A correctly identifies the maximum as (2,9) because the vertex form shows the peak at x=2, H(2)=9. Choice D identifies the maximum but at the wrong location: (-2,9) might come from misreading the vertex, but it's at x=2—make sure you're finding the right extreme! Quick reference for graph features: (1) Intercepts—where graph crosses axes, write as points (x, y), (2) Increasing/decreasing—read left to right, going up = increasing, going down = decreasing, state as x-intervals, (3) Positive/negative—above x-axis = positive, below = negative, state as x-intervals, (4) Maximum/minimum—highest/lowest points, give as points (x, y) or just y-value if asked, (5) End behavior—what happens at far left and far right of graph.

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For f(x) = -3x³ + 2x, the end behavior is determined by the leading term -3x³. Since this is a negative odd-degree term, as x → ∞ (x gets very large positive), -3x³ → -∞ (very large negative), and as x → -∞ (x gets very large negative), -3x³ → ∞ (very large positive). The +2x term doesn't affect end behavior because x³ grows much faster than x. Choice B correctly states 'As x→∞, f(x)→-∞ and as x→-∞, f(x)→∞' because the negative cubic function goes down on the right and up on the left. Choice A reverses this behavior—that would be true for a positive cubic like +3x³, not a negative one! End behavior shortcut for polynomials: look at the leading term (highest degree). Negative odd power (-x, -x³), left goes up, right goes down. This creates the characteristic 'S-shape' of cubic functions, just flipped when negative!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -(x-4)² + 9, this is a parabola in vertex form with vertex at (4, 9). Since the coefficient of the squared term is negative (-1), the parabola opens downward. This means the function increases as we approach the vertex from the left and decreases as we move away from the vertex to the right. Therefore, the function is increasing for all x-values less than 4, which we write as the interval (-∞, 4). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from the left until it reaches its maximum at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after reaching the maximum at x = 4, the profit drops as we sell more items, perhaps due to oversupply or increased costs. It's easy to confuse which side is increasing! For parabolas, remember: if it opens down (negative coefficient), it increases on the left of the vertex and decreases on the right. If it opens up (positive coefficient), it decreases on the left and increases on the right. The vertex is always the turning point where the behavior changes!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' In this context where the function models the height of a ball thrown upward over time, the y-intercept at (0, 15) means the initial height of the ball is 15 meters when t=0, before any time has passed. Choice B correctly identifies the y-intercept as (0,15) and interprets it as the initial height of the ball at t=0 because plugging t=0 into h(t) gives h(0)=15, representing the starting point. Choice A confuses the y-intercept with an x-intercept: (15,0) would be where height is zero, like when the ball hits the ground, but that's not the y-intercept—always check by setting the input to zero for y-intercept! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. The function has a minimum at (2, -3) and is symmetric about the vertical line x = 2. For a parabola or any function with vertical line symmetry, the axis of symmetry is the vertical line that acts as a mirror for the graph. Since we're told the function is symmetric about x = 2, this is precisely the axis of symmetry! Choice C correctly identifies the axis of symmetry as x = 2 because this is the vertical line about which the function is symmetric—it's the line that would fold the graph onto itself. Choice A gives y = 2, which would be a horizontal line—but axes of symmetry for functions are typically vertical lines of the form x = a, not horizontal lines! For functions with a vertex (like parabolas), the axis of symmetry always passes through the vertex. Since the minimum is at (2, -3), the axis of symmetry must be the vertical line x = 2. This makes sense: the graph is the same distance from this line on both sides!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Key features tell the story of a function: intercepts show where the function equals zero or starts, increasing/decreasing intervals show where it's rising or falling, maxima and minima show the peaks and valleys, and end behavior describes what happens as x gets very large or very small. Each feature reveals something important about the relationship being modeled. For end behavior of this exponential function N(t) = 200 · $(1.5)^t$, since the base 1.5 > 1, as t increases to infinity, N(t) grows without bound to infinity, modeling unlimited bacteria growth over time. Choice D correctly describes the end behavior as t → ∞, N(t) → ∞ because exponential growth with base >1 keeps multiplying and gets larger. Choice C misreads the growth: claiming N(t) → -∞ might confuse with decay, but since base >1, it's growth, not decay to negative—check the base! End behavior shortcut for polynomials: look at the leading term (highest degree). If it's positive even power (x², x⁴), both ends go up. Negative even power (-x², -x⁴), both ends go down. Positive odd power (x, x³), left goes down, right goes up. Negative odd power (-x, -x³), left goes up, right goes down. The leading term dominates for large |x|!

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. To identify where a function is increasing, look at the graph from left to right: if the graph is going upward (climbing), the function is increasing on that interval. If it's going downward (falling), it's decreasing. These intervals are described using the x-values, like 'increasing on (2, 5)' means as x goes from 2 to 5, the y-values are rising. For the profit function P(x) = -x² + 8x - 12, this is a downward-opening parabola (negative x² coefficient), which increases from the left up to its vertex, then decreases afterward. The vertex occurs at x = -b/(2a) = -8/(2(-1)) = 4, so the function increases on (-∞, 4) and decreases on (4, ∞). Choice B correctly identifies the increasing interval as (-∞, 4) because for a downward-opening parabola, the function rises from negative infinity up to the vertex at x = 4. Choice A gives (4, ∞), which is actually where the function is decreasing—after the peak at x = 4, the profit starts falling as we produce too many items. To avoid interval confusion: intervals ALWAYS use x-values (the inputs), never y-values! When we say 'increasing on (-∞, 4),' we mean 'as x goes from negative infinity to 4, y is rising.' For parabolas, remember: if it opens down, it increases then decreases; if it opens up, it decreases then increases.

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' To find x-intercepts of H(x) = -2x² + 12x + 5, we set H(x) = 0 and solve: -2x² + 12x + 5 = 0. Using the quadratic formula: x = [-12 ± √(144 - 4(-2)(5))] / (2(-2)) = [-12 ± √(144 + 40)] / (-4) = [-12 ± √184] / (-4) = [-12 ± 2√46] / (-4) ≈ 0.5 and 5.5. So the x-intercepts are approximately (0.5, 0) and (5.5, 0), representing where the roller coaster is at ground level. Choice C correctly identifies the x-intercepts as (0.5, 0) and (5.5, 0) because these are the points where the height function equals zero—where the roller coaster touches the ground. Choice A gives points with y-coordinate 5, but x-intercepts must have y-coordinate 0 by definition—they're where the graph crosses the x-axis, not some other horizontal line. Context interpretation: x-intercepts in height problems often mean 'when does the object reach ground level?' Here, the coaster starts underground, rises above ground at x = 0.5, and returns to ground at x = 5.5.

This question tests your ability to identify and interpret key features of functions from their graphs, tables, or formulas—features like intercepts, where the function increases or decreases, maximum and minimum values, and end behavior. Intercepts are special points: the y-intercept (0, b) is where the graph crosses the y-axis (this is the starting value when x = 0), and x-intercepts (a, 0) are where the graph crosses the x-axis (these are the zeros—where the function equals zero). In context, intercepts often have important meanings like 'initial value' or 'when does the quantity reach zero?' In this context where the function models population growth, the y-intercept at (0,500) means the initial population at t=0 is 500. Choice B correctly interprets the y-intercept as the initial population at t=0 because plugging t=0 gives P(0)=500. Choice A confuses y-intercept with growth factor: the 1.2 is the growth factor, while 500 is the starting value—remember, y-intercept is f(0)! Context interpretation trick: intercepts often mean 'starting value' (y-intercept) or 'when does it reach zero' (x-intercept). Maxima/minima often mean 'best/worst case' or 'peak/valley.' Increasing means 'getting better' or 'growing,' decreasing means 'getting worse' or 'shrinking.' Translate the math features into the context language!