Match the graphs with the conditions.

Matching Graphs to Conditions

This task involves matching different graphical representations of functions (parabolas and lines) to their corresponding mathematical conditions or equations. We will analyze each graph and its associated options to determine the correct pairings.

Section 1: Parabolas

The first set of graphs shows parabolas. We need to match them with the conditions related to the coefficients 'a' and 'c' in the quadratic equation $$y = ax^2 + bx + c$$. The shape and position of the parabola provide clues about these coefficients.

• Graph A (Parabola): This parabola opens upwards, which means the coefficient 'a' must be positive ($$a > 0$$). The vertex of the parabola is above the x-axis, and it intersects the y-axis at a positive value. If we consider the standard form $$y = ax^2 + c$$ (assuming b=0 for simplicity or looking at the vertex on the y-axis), the y-intercept is 'c'. Since the y-intercept is positive, $$c > 0$$. Therefore, the condition $$a > 0, c > 0$$ matches this graph. Looking at the options provided (1) $$a < 0, c > 0$$; (2) $$a > 0, c > 0$$; (3) $$a > 0, c < 0$$, option (2) seems to be the best fit.
• Graph Б (Parabola): This parabola opens downwards, indicating that 'a' is negative ($$a < 0$$). The vertex is above the x-axis, and the y-intercept is positive, so $$c > 0$$. This matches the condition $$a < 0, c > 0$$, which corresponds to option (1).
• Graph B (Parabola): This parabola opens upwards, so $$a > 0$$. The vertex is below the x-axis, and the y-intercept is negative, so $$c < 0$$. This matches the condition $$a > 0, c < 0$$, which corresponds to option (3).

Answer for Parabolas:

Section 2: Lines

The second set of graphs shows straight lines. We need to match them with the given equations of lines, where $$y = kx + b$$. The slope 'k' determines the steepness and direction of the line, and the y-intercept 'b' is where the line crosses the y-axis.

• Graph A (Line): This line has a positive slope (it rises from left to right) and a positive y-intercept. Let's examine the options: (1) $$y = 2x + 6$$, (2) $$y = -2x - 6$$, (3) $$y = -2x + 6$$. Equation (1) $$y = 2x + 6$$ has a positive slope ($$k=2$$) and a positive y-intercept ($$b=6$$). This matches Graph A.
• Graph Б (Line): This line has a negative slope (it falls from left to right) and a negative y-intercept. Examining the options, equation (2) $$y = -2x - 6$$ has a negative slope ($$k=-2$$) and a negative y-intercept ($$b=-6$$). This matches Graph Б.
• Graph B (Line): This line has a negative slope and a positive y-intercept. Equation (3) $$y = -2x + 6$$ has a negative slope ($$k=-2$$) and a positive y-intercept ($$b=6$$). This matches Graph B.

Answer for Lines:

Section 3: Lines (k and b conditions)

The third set of graphs also shows straight lines, this time with conditions on the slope 'k' and y-intercept 'b'.

• Graph A (Line): This line has a positive slope (rising from left to right) and a positive y-intercept. The conditions are: (1) $$k < 0, b < 0$$; (2) $$k < 0, b > 0$$; (3) $$k > 0, b > 0$$. The condition $$k > 0, b > 0$$ matches this graph. This corresponds to option (3).
• Graph Б (Line): This line has a negative slope (falling from left to right) and a positive y-intercept. This matches the condition $$k < 0, b > 0$$, which corresponds to option (2).
• Graph B (Line): This line has a negative slope and a negative y-intercept. This matches the condition $$k < 0, b < 0$$, which corresponds to option (1).

Answer for Lines (k, b conditions):