Question 1

How do you graph a linear equation y = mx + b?

Accepted Answer

Step 1: Plot the y-intercept (0, b). Step 2: Use the slope to find a second point: from (0, b), move right 1 unit and up m units (or down |m| units if m < 0) to reach the point (1, b+m). Step 3: Draw a straight line through the two points and extend it in both directions. Alternative: find the x-intercept by setting y = 0 → x = −b/m, plot (−b/m, 0), and draw the line through both intercepts.

Question 2

How do you find the vertex of a parabola y = ax² + bx + c?

Accepted Answer

The x-coordinate of the vertex is h = −b/(2a). The y-coordinate is k = c − b²/(4a), or equivalently, substitute h back: k = a(h²) + b(h) + c. Example: y = 2x² − 8x + 3. h = −(−8)/(2×2) = 8/4 = 2. k = 2(4) − 8(2) + 3 = 8 − 16 + 3 = −5. Vertex: (2, −5). Since a = 2 > 0, the parabola opens upward and this is the minimum point.

Question 3

How do you find x-intercepts of a graph?

Accepted Answer

The x-intercepts are points where y = 0 — where the graph crosses the x-axis. For a linear equation y = mx + b: set 0 = mx + b → x = −b/m (one intercept). For a quadratic y = ax² + bx + c: set 0 = ax² + bx + c and use the quadratic formula x = (−b ± √(b²−4ac))/(2a). The discriminant b²−4ac tells you how many x-intercepts: positive → 2, zero → 1 (touches), negative → 0 (does not cross x-axis).

Question 4

What does the slope tell you about the line?

Accepted Answer

Slope m = rise/run = (y₂−y₁)/(x₂−x₁). Interpretation: m = 2 means for every 1 unit right, the line rises 2 units. m = −1/3 means for every 3 units right, the line falls 1 unit. m = 0 is a horizontal line. Undefined slope is a vertical line. Positive slope: line rises left to right. Negative slope: line falls left to right. Steeper lines have larger |m|. Parallel lines have equal slopes. Perpendicular lines have slopes whose product is −1 (negative reciprocals).

Question 5

How does a graph change when you change a, b, or c in y = ax² + bx + c?

Accepted Answer

Parameter a: controls width and direction. |a| > 1 narrows the parabola; 0 < |a| < 1 widens it; a < 0 flips it downward. Parameter c: shifts the parabola vertically (the y-intercept is always c). Parameter b: shifts the vertex horizontally — increasing b moves the axis of symmetry x = −b/(2a) left or right. Vertex form y = a(x−h)² + k makes the shifts explicit: h is the horizontal shift and k is the vertical shift from the origin.

Feature	Linear y=mx+b	Quadratic y=ax²+bx+c
y-intercept	(0, b)	(0, c)
x-intercept(s)	(−b/m, 0) — one	Use quadratic formula — 0, 1, or 2
Shape	Straight line	Parabola (U-shaped)
Vertex/Max/Min	No vertex	(−b/2a, c−b²/4a)
Axis of symmetry	None	x = −b/(2a) — vertical line
Domain	(−∞, ∞)	(−∞, ∞)
Range	(−∞, ∞)	[k, ∞) if a>0; (−∞, k] if a<0
Slope meaning	Rise/run, rate of change	Not constant — rate changes

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