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Quiz: Graphing and Key Features

Test your understanding of graphs, intercepts, turning points, concavity, and graphing technology with these review questions.

1. What is the \(y\)-intercept of \(f(x) = 2x^2 - 5x + 7\)?

  1. \((0, 7)\)
  2. \((7, 0)\)
  3. \((0, -5)\)
  4. \((0, 2)\)
Show Answer

The correct answer is A. To find the \(y\)-intercept, substitute \(x = 0\) into the function: \(f(0) = 2(0)^2 - 5(0) + 7 = 7\). The \(y\)-intercept is always a point on the \(y\)-axis, so its \(x\)-coordinate is \(0\), giving \((0, 7)\). A common mistake is to confuse the \(y\)-intercept with the \(x\)-intercept, which is where the graph crosses the \(x\)-axis.

Concept Tested: Y-Intercept

2. Which term refers to the input values where a function equals zero?

  1. Turning points
  2. Maxima
  3. Zeros
  4. Inflection points
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The correct answer is C. The zeros of a function are the values of \(x\) where \(f(x) = 0\). These are also called roots (especially when solving equations) or \(x\)-intercepts (when viewed as points on the graph). Turning points mark where the graph changes direction, maxima are peaks, and inflection points mark where concavity changes — none of these require the output to be zero.

Concept Tested: Zeros of a Function

3. What is the slope (gradient) of the line through the points \((2, 1)\) and \((5, 10)\)?

  1. \(\frac{1}{3}\)
  2. \(9\)
  3. \(\frac{9}{7}\)
  4. \(3\)
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The correct answer is D. The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting: \(m = \frac{10 - 1}{5 - 2} = \frac{9}{3} = 3\). A common mistake is to invert the formula (giving \(\frac{1}{3}\)) or to subtract the coordinates in inconsistent orders. Always match the subtraction order in the numerator and denominator.

Concept Tested: Slope

4. A parabola opens downward and has its turning point at \((2, 5)\). This turning point is:

  1. A minimum point
  2. A maximum point
  3. An inflection point
  4. A zero
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The correct answer is B. When a parabola opens downward, its graph rises to a peak and then falls, so the turning point is the highest value of the function — a maximum. A minimum would occur if the parabola opened upward. Inflection points mark changes in concavity (not applicable to parabolas, which have constant concavity), and zeros are where the function equals zero.

Concept Tested: Maximum Point

5. On which interval is \(f(x) = x^2\) decreasing?

  1. \((0, \infty)\)
  2. \((-\infty, 0)\)
  3. \((-\infty, \infty)\)
  4. \([0, \infty)\)
Show Answer

The correct answer is B. Looking at the parabola \(y = x^2\), as \(x\) moves from \(-\infty\) toward \(0\), the output values decrease from very large positive numbers down to \(0\). Once past \(x = 0\), the function begins increasing. So it is decreasing on \((-\infty, 0)\) and increasing on \((0, \infty)\). The turning point at \(x = 0\) separates the two behaviors.

Concept Tested: Decreasing Function

6. A graph that curves like a bowl (holding water) is said to be:

  1. Concave down
  2. Increasing
  3. Concave up
  4. Linear
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The correct answer is C. A concave-up graph curves upward like a bowl or the letter U — it would hold water poured onto it. Concave-down graphs curve like an umbrella and shed water. Note that concavity is independent of whether the function is increasing or decreasing: a function can be increasing and concave down, decreasing and concave up, and so on.

Concept Tested: Concavity

7. Find the intersection points of \(f(x) = x^2\) and \(g(x) = x + 2\).

  1. \((2, 4)\) and \((-1, 1)\)
  2. \((1, 1)\) and \((-2, 4)\)
  3. \((2, 4)\) only
  4. \((0, 0)\) and \((2, 4)\)
Show Answer

The correct answer is A. Set the functions equal: \(x^2 = x + 2\), giving \(x^2 - x - 2 = 0\). Factor: \((x - 2)(x + 1) = 0\), so \(x = 2\) or \(x = -1\). Substitute back: when \(x = 2\), \(y = 4\); when \(x = -1\), \(y = 1\). The intersection points are \((2, 4)\) and \((-1, 1)\). Always check by verifying both functions give the same output at each point.

Concept Tested: Intersection of Curves

8. An IB exam question says "sketch the graph of \(f(x)\)." This instruction means you should:

  1. Use graph paper to plot exact points to scale
  2. Use a calculator to produce a printout
  3. Produce a quick freehand diagram with labeled key features
  4. Describe the graph in words only
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The correct answer is C. In IB Mathematics, "sketch" means produce a freehand diagram showing the essential shape and key features (intercepts, turning points, asymptotes) without needing to be drawn to scale. "Draw" would require precise, scaled work. Sketches should still have labeled features to earn full marks — they are approximate in scale but accurate in shape and structure.

Concept Tested: Sketching vs Drawing

9. Round \(0.041728\) to 3 significant figures.

  1. \(0.041\)
  2. \(0.0417\)
  3. \(0.042\)
  4. \(0.0420\)
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The correct answer is B. Significant figures start with the first non-zero digit. In \(0.041728\), the leading zeros are not significant. The first significant digit is \(4\), the second is \(1\), the third is \(7\). Looking at the next digit (\(2\)) to decide on rounding: \(2 < 5\), so round down, giving \(0.0417\). Don't confuse significant figures with decimal places — these count different things.

Concept Tested: Significant Figures

10. An inflection point is a point where:

  1. The graph crosses the \(x\)-axis
  2. The function reaches a maximum or minimum
  3. The slope becomes zero
  4. The concavity changes
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The correct answer is D. An inflection point is where a graph changes its concavity — from concave up to concave down, or vice versa. At this point, the curvature reverses direction. Option A describes an \(x\)-intercept, option B describes a turning point, and option D may describe either a turning point or a stationary inflection point. Concavity change is the defining property.

Concept Tested: Inflection Point