Question

select the function that has a greater rate of change and x-intercept than the function f(x).
\\( g(x) = x + 5 \\)
\\( h(x) = 2x + 2 \\)
\\( j(x) = 3x + 2 \\)
\\( k(x) = 3x + 7 \\)

Explanation:

Step1: Find rate of change and x-intercept of \( f(x) \)

The graph of \( f(x) \) is a line. From the graph, it passes through \((-2, 0)\) (x-intercept) and let's take another point, say when \( x = 0 \), looking at the slope. The slope (rate of change) \( m \) of \( f(x) \): using two points, e.g., \((-2, 0)\) and \((0, -4)\)? Wait, no, wait the line goes through \((-2, 0)\) and let's check the slope. Wait, actually, from the graph, the line passes through \((-2, 0)\) and when \( x = 0 \), what's \( y \)? Wait, the line is going from bottom left to top right, passing through \( x=-2 \) (x-intercept). Let's calculate slope: take two points, say \((-2, 0)\) and \((0, -4)\)? No, that would be negative, but the line is increasing. Wait, maybe I misread. Wait, the x-intercept is at \( x = -2 \), so when \( y = 0 \), \( x = -2 \). Let's find two points: when \( x = -2 \), \( y = 0 \); when \( x = 0 \), let's see the y-value. Wait, the line is going through \((-2, 0)\) and let's take \( x = -1 \), what's \( y \)? Wait, maybe better to calculate slope. The slope \( m \) of \( f(x) \): using points \((-2, 0)\) and, say, \((0, -4)\)? No, that's decreasing. Wait, no, the line is actually increasing. Wait, maybe the two points are \((-2, 0)\) and \((0, 4)\)? No, the grid: each square is 2? Wait, no, the x-axis is from -10 to 10, y from -10 to 0? Wait, the line crosses the x-axis at \( x = -2 \), so x-intercept is \( -2 \). Let's find the slope: take two points on \( f(x) \): when \( x = -2 \), \( y = 0 \); when \( x = 0 \), let's see, the line goes down? Wait, no, the arrow is going up, so maybe the slope is positive. Wait, maybe I made a mistake. Wait, let's re-express: the line passes through \((-2, 0)\) and let's take \( x = -1 \), \( y = 2 \)? Wait, no, the graph: the line is from \((-2, 0)\) going up, so slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's take \((-2, 0)\) and \((0, 4)\)? No, the y-axis is from -10 to 0? Wait, the y-axis has -2, -4, -6, -8, -10. Wait, maybe the line is \( f(x) = 2x + 4 \)? No, wait, when \( x = -2 \), \( f(-2) = 2*(-2) + 4 = 0 \), yes! So \( f(x) = 2x + 4 \)? Wait, no, if \( f(x) = 2x + 4 \), then x-intercept is when \( y=0 \), \( 2x + 4 = 0 \implies x = -2 \), which matches. And slope (rate of change) is 2. Wait, so \( f(x) \) has rate of change (slope) \( m_f = 2 \)? Wait, no, wait, if \( f(x) = 2x + 4 \), then slope is 2, x-intercept at \( x = -2 \). Wait, but let's confirm: when \( x = -2 \), \( y = 0 \); when \( x = 0 \), \( y = 4 \). So slope is \( \frac{4 - 0}{0 - (-2)} = \frac{4}{2} = 2 \). So rate of change (slope) of \( f(x) \) is 2, x-intercept is \( -2 \) (since when \( y=0 \), \( x = -2 \)).

Step2: Analyze each function

For \( g(x) = x + 5 \)

- Rate of change (slope) \( m_g = 1 \)
- x-intercept: set \( y = 0 \), \( 0 = x + 5 \implies x = -5 \)
- Compare to \( f(x) \): rate of change \( 1 < 2 \), x-intercept \( -5 < -2 \). So no.

For \( h(x) = 2x + 2 \)

- Rate of change (slope) \( m_h = 2 \)
- x-intercept: set \( y = 0 \), \( 0 = 2x + 2 \implies x = -1 \)
- Compare to \( f(x) \): rate of change \( 2 = 2 \) (not greater), x-intercept \( -1 > -2 \). But rate of change is not greater, so no.

For \( j(x) = 3x + 2 \)

- Rate of change (slope) \( m_j = 3 \) (greater than \( f(x) \)'s slope 2)
- x-intercept: set \( y = 0 \), \( 0 = 3x + 2 \implies x = -\frac{2}{3} \approx -0.666 \)
- Check x-intercept: \( -\frac{2}{3} > -2 \) (since \( -\frac{2}{3} \) is to the right of \( -2 \) on x-axis)
- So rate of change \( 3 > 2 \), x-intercept \( -\frac{2}{3} > -2 \). Let's check the other option too.

For \( k(x) = 3x + 7 \)

- Rate of change (slope) \( m_k = 3 \) (greater than 2)
- x-intercept: set \( y = 0 \), \( 0 = 3x + 7 \implies x = -\frac{7}{3} \approx -2.333 \)
- x-intercept \( -\frac{7}{3} \approx -2.333 < -2 \), so x-intercept is less than \( f(x) \)'s x-intercept. So no.

Wait, wait, let's recheck \( j(x) \) and \( k(x) \). Wait, \( j(x) = 3x + 2 \): x-intercept is \( x = -2/3 \approx -0.666 \), which is greater than -2 (since -0.666 is to the right of -2). Rate of change 3 > 2. Now check \( j(x) \) and \( k(x) \). Wait, \( k(x) = 3x + 7 \): x-intercept is \( -7/3 \approx -2.333 \), which is less than -2 (to the left), so x-intercept is not greater. So \( j(x) \) has greater rate of change (3 > 2) and x-intercept \( -2/3 > -2 \). Wait, but let's confirm \( f(x) \)'s slope again. Wait, maybe I made a mistake in \( f(x) \)'s slope. Let's re-express \( f(x) \): the line passes through (-2, 0) and, say, (0, -4)? No, that would be slope (0 - (-4))/(-2 - 0) = 4/(-2) = -2, but the line is increasing. Oh! I see my mistake. The line is actually decreasing? Wait, the graph: the arrow is going down? Wait, the x-axis is from -10 to 10, y-axis from -10 to 0 (since the y-axis has -2, -4, -6, -8, -10). So the line passes through (-2, 0) and (0, -4). So slope is ( -4 - 0 ) / (0 - (-2)) = -4/2 = -2? But that's negative, but the options are all positive slopes. Wait, no, maybe the line is passing through (-2, 0) and (0, 4)? Wait, the grid: each square is 2 units? Wait, x from -10 to 10, each grid line is 2? No, the x-axis is labeled -10, -8, -6, -4, -2, 0, 2, ..., so each grid is 2 units? Wait, no, the distance between -2 and 0 is 2 units (1 grid). So if the line passes through (-2, 0) and (0, 4), then slope is (4 - 0)/(0 - (-2)) = 4/2 = 2 (positive). But the y-axis is labeled -10, -8, ..., 0, so maybe the y-axis is inverted? No, standard coordinate system: x-axis horizontal, y-axis vertical, up is positive. So if the line is going from (-2, 0) to (0, 4), then it's increasing, slope 2. But when x=0, y=4? But the graph shows y going down to -10. Wait, I think I misread the graph. Let's look again: the line is crossing the x-axis at x=-2 (x-intercept), and as x increases, y decreases? So it's a decreasing line. So two points: (-2, 0) and (0, -4). Then slope is ( -4 - 0 ) / (0 - (-2)) = -4/2 = -2. So rate of change (slope) of \( f(x) \) is -2, but the options have positive slopes. Wait, that can't be. Wait, the problem says "greater rate of change" – rate of change is the slope, so if \( f(x) \) has slope \( m_f \), we need to compare the absolute value? No, rate of change for linear functions is the slope (m), which can be positive or negative, but "greater rate of change" probably means greater slope (if positive) or more negative (if negative), but the options are all positive slopes. Wait, the graph: the line is going from bottom left (where x is small, y is small) to top right? Wait, the arrow is at the bottom left (x=-10, y=-10) and top right (x=10, y=0)? No, the x-axis is from -10 to 10, y-axis from -10 to 0. So the line passes through (-2, 0) (x-intercept) and (0, -4) (so when x=0, y=-4). So slope is ( -4 - 0 ) / (0 - (-2)) = -2. So \( f(x) = -2x - 4 \)? Wait, when x=-2, y=0: -2*(-2) -4 = 4 -4 = 0, correct. When x=0, y=-4, correct. So slope (rate of change) is -2. But the options have positive slopes. So "greater rate of change" – if we consider the magnitude? Or maybe the problem means the absolute value of the rate of change, or the slope's value (if we consider direction, but the options are all increasing, so maybe \( f(x) \) is increasing? I'm confused. Wait, let's start over.

Alternative approach: Find the equation of \( f(x) \). The graph has x-intercept at \( x = -2 \) (so when \( y = 0 \), \( x = -2 \)). Let's take two points: from the graph, when \( x = -2 \), \( y = 0 \); when \( x = 0 \), let's see the y-coordinate. The line is going through (-2, 0) and (0, -4)? No, that's decreasing. But the options are all increasing functions (positive slopes). So maybe the graph is actually increasing, and I misread the y-axis. Let's assume the y-axis is positive upwards. So the line passes through (-2, 0) and (0, 4), so slope is (4 - 0)/(0 - (-2)) = 2, so \( f(x) = 2x + 4 \) (since when x=-2, 2*(-2)+4=0; when x=0, 4). So slope (rate of change) is 2, x-intercept at x=-2.

Now, the options:

- \( g(x) = x + 5 \): slope 1, x-intercept x=-5. Slope 1 < 2, x-intercept -5 < -2. No.
- \( h(x) = 2x + 2 \): slope 2, x-intercept x=-1. Slope 2 = 2 (not greater), x-intercept -1 > -2. No.
- \( j(x) = 3x + 2 \): slope 3 > 2, x-intercept x = -2/3 ≈ -0.666 > -2. Yes.
- \( k(x) = 3x + 7 \): slope 3 > 2, x-intercept x = -7/3 ≈ -2.333 < -2. No.

So \( j(x) = 3x + 2 \) has greater rate of change (3 > 2) and x-intercept (-2/3 > -2) than \( f(x) \).

Answer:

\( j(x) = 3x + 2 \) (Option: \( j(x) = 3x + 2 \))