sat suite question viewer

Advanced Math / Nonlinear functions Difficulty: Hard

The function $f$ is defined by $f (x) = a^{x} + b$ , where $a$ and $b$ are constants and $a greater than 0$ . In the xy-plane, the graph of $y = f (x)$ has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ and passes through the point $left parenthesis 2 comma 23 right parenthesis$ . What is the value of $a + b$ ?

Back question 195 of 229 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229

Explanation

The correct answer is $negative 19$ . It's given that function $f$ is defined by $f (x) = a^{x} + b$ , where $a$ and $b$ are constants and $a greater than 0$ . It's also given that the graph of $y = f (x)$ in the xy-plane has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ and passes through the point $left parenthesis 2 comma 23 right parenthesis$ . Since the graph has a y-intercept at $left parenthesis 0 comma negative 25 right parenthesis$ , $f (0) = - 25$ . Substituting $0$ for $x$ in the given equation yields $f (0) = a^{0} + b$ , or $f (0) = 1 + b$ , and substituting $negative 25$ for $f left parenthesis 0 right parenthesis$ in this equation yields $- 25 = 1 + b$ . Subtracting $1$ from each side of this equation yields $- 26 = b$ . Substituting $negative 26$ for $b$ in the equation $f (x) = a^{x} + b$ yields $f (x) = a^{x} - 26$ . Since the graph also passes through the point $left parenthesis 2 comma 23 right parenthesis$ , $f left parenthesis 2 right parenthesis equals 23$ . Substituting $2$ for $x$ in the equation $f (x) = a^{x} - 26$ yields $f (2) = a^{2} - 26$ , and substituting $23$ for $f left parenthesis 2 right parenthesis$ yields $23 = a^{2} - 26$ . Adding $26$ to each side of this equation yields $49 equals a squared$ . Taking the square root of both sides of this equation yields $\pm 7 = a$ . Since it's given that $a greater than 0$ , the value of $a$ is $7$ . It follows that the value of $a + b$ is $7 minus 26$ , or $negative 19$ .